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    * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
    *
    * This code is free software; you can redistribute it and/or modify it
    * under the terms of the GNU General Public License version 2 only, as
    * published by the Free Software Foundation.  Oracle designates this
    * particular file as subject to the "Classpath" exception as provided
    * by Oracle in the LICENSE file that accompanied this code.
   *
   * This code is distributed in the hope that it will be useful, but WITHOUT
   * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
   * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
   * version 2 for more details (a copy is included in the LICENSE file that
   * accompanied this code).
   *
   * You should have received a copy of the GNU General Public License version
   * 2 along with this work; if not, write to the Free Software Foundation,
   * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
   *
   * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
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   */
  
  /*
   * Portions Copyright (c) 1995  Colin Plumb.  All rights reserved.
   */
  
  package java.math;
  
  import java.util.Random;
  import java.io.*;
  
  /**
   * Immutable arbitrary-precision integers.  All operations behave as if
   * BigIntegers were represented in two's-complement notation (like Java's
   * primitive integer types).  BigInteger provides analogues to all of Java's
   * primitive integer operators, and all relevant methods from java.lang.Math.
   * Additionally, BigInteger provides operations for modular arithmetic, GCD
   * calculation, primality testing, prime generation, bit manipulation,
   * and a few other miscellaneous operations.
   *
   * <p>Semantics of arithmetic operations exactly mimic those of Java's integer
   * arithmetic operators, as defined in <i>The Java Language Specification</i>.
   * For example, division by zero throws an {@code ArithmeticException}, and
   * division of a negative by a positive yields a negative (or zero) remainder.
   * All of the details in the Spec concerning overflow are ignored, as
   * BigIntegers are made as large as necessary to accommodate the results of an
   * operation.
   *
   * <p>Semantics of shift operations extend those of Java's shift operators
   * to allow for negative shift distances.  A right-shift with a negative
   * shift distance results in a left shift, and vice-versa.  The unsigned
   * right shift operator ({@code >>>}) is omitted, as this operation makes
   * little sense in combination with the "infinite word size" abstraction
   * provided by this class.
   *
   * <p>Semantics of bitwise logical operations exactly mimic those of Java's
   * bitwise integer operators.  The binary operators ({@code and},
   * {@code or}, {@code xor}) implicitly perform sign extension on the shorter
   * of the two operands prior to performing the operation.
   *
   * <p>Comparison operations perform signed integer comparisons, analogous to
   * those performed by Java's relational and equality operators.
   *
   * <p>Modular arithmetic operations are provided to compute residues, perform
   * exponentiation, and compute multiplicative inverses.  These methods always
   * return a non-negative result, between {@code 0} and {@code (modulus - 1)},
   * inclusive.
   *
   * <p>Bit operations operate on a single bit of the two's-complement
   * representation of their operand.  If necessary, the operand is sign-
   * extended so that it contains the designated bit.  None of the single-bit
   * operations can produce a BigInteger with a different sign from the
   * BigInteger being operated on, as they affect only a single bit, and the
   * "infinite word size" abstraction provided by this class ensures that there
   * are infinitely many "virtual sign bits" preceding each BigInteger.
   *
   * <p>For the sake of brevity and clarity, pseudo-code is used throughout the
   * descriptions of BigInteger methods.  The pseudo-code expression
   * {@code (i + j)} is shorthand for "a BigInteger whose value is
   * that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
   * The pseudo-code expression {@code (i == j)} is shorthand for
   * "{@code true} if and only if the BigInteger {@code i} represents the same
   * value as the BigInteger {@code j}."  Other pseudo-code expressions are
   * interpreted similarly.
   *
   * <p>All methods and constructors in this class throw
   * {@code NullPointerException} when passed
   * a null object reference for any input parameter.
   *
   * @see     BigDecimal
   * @author  Josh Bloch
   * @author  Michael McCloskey
   * @since JDK1.1
   */
  
  public class BigInteger extends Number implements Comparable<BigInteger> {
     /**
      * The signum of this BigInteger: -1 for negative, 0 for zero, or
      * 1 for positive.  Note that the BigInteger zero <i>must</i> have
      * a signum of 0.  This is necessary to ensures that there is exactly one
      * representation for each BigInteger value.
      *
      * @serial
      */
     final int signum;
 
     /**
      * The magnitude of this BigInteger, in <i>big-endian</i> order: the
      * zeroth element of this array is the most-significant int of the
      * magnitude.  The magnitude must be "minimal" in that the most-significant
      * int ({@code mag[0]}) must be non-zero.  This is necessary to
      * ensure that there is exactly one representation for each BigInteger
      * value.  Note that this implies that the BigInteger zero has a
      * zero-length mag array.
      */
     final int[] mag;
 
     // These "redundant fields" are initialized with recognizable nonsense
     // values, and cached the first time they are needed (or never, if they
     // aren't needed).
 
      /**
      * One plus the bitCount of this BigInteger. Zeros means unitialized.
      *
      * @serial
      * @see #bitCount
      * @deprecated Deprecated since logical value is offset from stored
      * value and correction factor is applied in accessor method.
      */
     @Deprecated
     private int bitCount;
 
     /**
      * One plus the bitLength of this BigInteger. Zeros means unitialized.
      * (either value is acceptable).
      *
      * @serial
      * @see #bitLength()
      * @deprecated Deprecated since logical value is offset from stored
      * value and correction factor is applied in accessor method.
      */
     @Deprecated
     private int bitLength;
 
     /**
      * Two plus the lowest set bit of this BigInteger, as returned by
      * getLowestSetBit().
      *
      * @serial
      * @see #getLowestSetBit
      * @deprecated Deprecated since logical value is offset from stored
      * value and correction factor is applied in accessor method.
      */
     @Deprecated
     private int lowestSetBit;
 
     /**
      * Two plus the index of the lowest-order int in the magnitude of this
      * BigInteger that contains a nonzero int, or -2 (either value is acceptable).
      * The least significant int has int-number 0, the next int in order of
      * increasing significance has int-number 1, and so forth.
      * @deprecated Deprecated since logical value is offset from stored
      * value and correction factor is applied in accessor method.
      */
     @Deprecated
     private int firstNonzeroIntNum;
 
     /**
      * This mask is used to obtain the value of an int as if it were unsigned.
      */
     final static long LONG_MASK = 0xffffffffL;
 
     //Constructors
 
     /**
      * Translates a byte array containing the two's-complement binary
      * representation of a BigInteger into a BigInteger.  The input array is
      * assumed to be in <i>big-endian</i> byte-order: the most significant
      * byte is in the zeroth element.
      *
      * @param  val big-endian two's-complement binary representation of
      *         BigInteger.
      * @throws NumberFormatException {@code val} is zero bytes long.
      */
     public BigInteger(byte[] val) {
         if (val.length == 0)
             throw new NumberFormatException("Zero length BigInteger");
 
         if (val[0] < 0) {
             mag = makePositive(val);
             signum = -1;
         } else {
             mag = stripLeadingZeroBytes(val);
             signum = (mag.length == 0 ? 0 : 1);
         }
     }
 
     /**
      * This private constructor translates an int array containing the
      * two's-complement binary representation of a BigInteger into a
      * BigInteger. The input array is assumed to be in <i>big-endian</i>
      * int-order: the most significant int is in the zeroth element.
      */
     private BigInteger(int[] val) {
         if (val.length == 0)
             throw new NumberFormatException("Zero length BigInteger");
 
         if (val[0] < 0) {
             mag = makePositive(val);
             signum = -1;
         } else {
             mag = trustedStripLeadingZeroInts(val);
             signum = (mag.length == 0 ? 0 : 1);
         }
     }
 
     /**
      * Translates the sign-magnitude representation of a BigInteger into a
      * BigInteger.  The sign is represented as an integer signum value: -1 for
      * negative, 0 for zero, or 1 for positive.  The magnitude is a byte array
      * in <i>big-endian</i> byte-order: the most significant byte is in the
      * zeroth element.  A zero-length magnitude array is permissible, and will
      * result in a BigInteger value of 0, whether signum is -1, 0 or 1.
      *
      * @param  signum signum of the number (-1 for negative, 0 for zero, 1
      *         for positive).
      * @param  magnitude big-endian binary representation of the magnitude of
      *         the number.
      * @throws NumberFormatException {@code signum} is not one of the three
      *         legal values (-1, 0, and 1), or {@code signum} is 0 and
      *         {@code magnitude} contains one or more non-zero bytes.
      */
     public BigInteger(int signum, byte[] magnitude) {
         this.mag = stripLeadingZeroBytes(magnitude);
 
         if (signum < -1 || signum > 1)
             throw(new NumberFormatException("Invalid signum value"));
 
         if (this.mag.length==0) {
             this.signum = 0;
         } else {
             if (signum == 0)
                 throw(new NumberFormatException("signum-magnitude mismatch"));
             this.signum = signum;
         }
     }
 
     /**
      * A constructor for internal use that translates the sign-magnitude
      * representation of a BigInteger into a BigInteger. It checks the
      * arguments and copies the magnitude so this constructor would be
      * safe for external use.
      */
     private BigInteger(int signum, int[] magnitude) {
         this.mag = stripLeadingZeroInts(magnitude);
 
         if (signum < -1 || signum > 1)
             throw(new NumberFormatException("Invalid signum value"));
 
         if (this.mag.length==0) {
             this.signum = 0;
         } else {
             if (signum == 0)
                 throw(new NumberFormatException("signum-magnitude mismatch"));
             this.signum = signum;
         }
     }
 
     /**
      * Translates the String representation of a BigInteger in the
      * specified radix into a BigInteger.  The String representation
      * consists of an optional minus or plus sign followed by a
      * sequence of one or more digits in the specified radix.  The
      * character-to-digit mapping is provided by {@code
      * Character.digit}.  The String may not contain any extraneous
      * characters (whitespace, for example).
      *
      * @param val String representation of BigInteger.
      * @param radix radix to be used in interpreting {@code val}.
      * @throws NumberFormatException {@code val} is not a valid representation
      *         of a BigInteger in the specified radix, or {@code radix} is
      *         outside the range from {@link Character#MIN_RADIX} to
      *         {@link Character#MAX_RADIX}, inclusive.
      * @see    Character#digit
      */
     public BigInteger(String val, int radix) {
         int cursor = 0, numDigits;
         final int len = val.length();
 
         if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
             throw new NumberFormatException("Radix out of range");
         if (len == 0)
             throw new NumberFormatException("Zero length BigInteger");
 
         // Check for at most one leading sign
         int sign = 1;
         int index1 = val.lastIndexOf('-');
         int index2 = val.lastIndexOf('+');
         if ((index1 + index2) <= -1) {
             // No leading sign character or at most one leading sign character
             if (index1 == 0 || index2 == 0) {
                 cursor = 1;
                 if (len == 1)
                     throw new NumberFormatException("Zero length BigInteger");
             }
             if (index1 == 0)
                 sign = -1;
         } else
             throw new NumberFormatException("Illegal embedded sign character");
 
         // Skip leading zeros and compute number of digits in magnitude
         while (cursor < len &&
                Character.digit(val.charAt(cursor), radix) == 0)
             cursor++;
         if (cursor == len) {
             signum = 0;
             mag = ZERO.mag;
             return;
         }
 
         numDigits = len - cursor;
         signum = sign;
 
         // Pre-allocate array of expected size. May be too large but can
         // never be too small. Typically exact.
         int numBits = (int)(((numDigits * bitsPerDigit[radix]) >>> 10) + 1);
         int numWords = (numBits + 31) >>> 5;
         int[] magnitude = new int[numWords];
 
         // Process first (potentially short) digit group
         int firstGroupLen = numDigits % digitsPerInt[radix];
         if (firstGroupLen == 0)
             firstGroupLen = digitsPerInt[radix];
         String group = val.substring(cursor, cursor += firstGroupLen);
         magnitude[numWords - 1] = Integer.parseInt(group, radix);
         if (magnitude[numWords - 1] < 0)
             throw new NumberFormatException("Illegal digit");
 
         // Process remaining digit groups
         int superRadix = intRadix[radix];
         int groupVal = 0;
         while (cursor < len) {
             group = val.substring(cursor, cursor += digitsPerInt[radix]);
             groupVal = Integer.parseInt(group, radix);
             if (groupVal < 0)
                 throw new NumberFormatException("Illegal digit");
             destructiveMulAdd(magnitude, superRadix, groupVal);
         }
         // Required for cases where the array was overallocated.
         mag = trustedStripLeadingZeroInts(magnitude);
     }
 
     // Constructs a new BigInteger using a char array with radix=10
     BigInteger(char[] val) {
         int cursor = 0, numDigits;
         int len = val.length;
 
         // Check for leading minus sign
         int sign = 1;
         if (val[0] == '-') {
             if (len == 1)
                 throw new NumberFormatException("Zero length BigInteger");
             sign = -1;
             cursor = 1;
         } else if (val[0] == '+') {
             if (len == 1)
                 throw new NumberFormatException("Zero length BigInteger");
             cursor = 1;
         }
 
         // Skip leading zeros and compute number of digits in magnitude
         while (cursor < len && Character.digit(val[cursor], 10) == 0)
             cursor++;
         if (cursor == len) {
             signum = 0;
             mag = ZERO.mag;
             return;
         }
 
         numDigits = len - cursor;
         signum = sign;
 
         // Pre-allocate array of expected size
         int numWords;
         if (len < 10) {
             numWords = 1;
         } else {
             int numBits = (int)(((numDigits * bitsPerDigit[10]) >>> 10) + 1);
             numWords = (numBits + 31) >>> 5;
         }
         int[] magnitude = new int[numWords];
 
         // Process first (potentially short) digit group
         int firstGroupLen = numDigits % digitsPerInt[10];
         if (firstGroupLen == 0)
             firstGroupLen = digitsPerInt[10];
         magnitude[numWords - 1] = parseInt(val, cursor,  cursor += firstGroupLen);
 
         // Process remaining digit groups
         while (cursor < len) {
             int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
             destructiveMulAdd(magnitude, intRadix[10], groupVal);
         }
         mag = trustedStripLeadingZeroInts(magnitude);
     }
 
     // Create an integer with the digits between the two indexes
     // Assumes start < end. The result may be negative, but it
     // is to be treated as an unsigned value.
     private int parseInt(char[] source, int start, int end) {
         int result = Character.digit(source[start++], 10);
         if (result == -1)
             throw new NumberFormatException(new String(source));
 
         for (int index = start; index<end; index++) {
             int nextVal = Character.digit(source[index], 10);
             if (nextVal == -1)
                 throw new NumberFormatException(new String(source));
             result = 10*result + nextVal;
         }
 
         return result;
     }
 
     // bitsPerDigit in the given radix times 1024
     // Rounded up to avoid underallocation.
     private static long bitsPerDigit[] = { 0, 0,
         1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
         3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
         4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
                                            5253, 5295};
 
     // Multiply x array times word y in place, and add word z
     private static void destructiveMulAdd(int[] x, int y, int z) {
         // Perform the multiplication word by word
         long ylong = y & LONG_MASK;
         long zlong = z & LONG_MASK;
         int len = x.length;
 
         long product = 0;
         long carry = 0;
         for (int i = len-1; i >= 0; i--) {
             product = ylong * (x[i] & LONG_MASK) + carry;
             x[i] = (int)product;
             carry = product >>> 32;
         }
 
         // Perform the addition
         long sum = (x[len-1] & LONG_MASK) + zlong;
         x[len-1] = (int)sum;
         carry = sum >>> 32;
         for (int i = len-2; i >= 0; i--) {
             sum = (x[i] & LONG_MASK) + carry;
             x[i] = (int)sum;
             carry = sum >>> 32;
         }
     }
 
     /**
      * Translates the decimal String representation of a BigInteger into a
      * BigInteger.  The String representation consists of an optional minus
      * sign followed by a sequence of one or more decimal digits.  The
      * character-to-digit mapping is provided by {@code Character.digit}.
      * The String may not contain any extraneous characters (whitespace, for
      * example).
      *
      * @param val decimal String representation of BigInteger.
      * @throws NumberFormatException {@code val} is not a valid representation
      *         of a BigInteger.
      * @see    Character#digit
      */
     public BigInteger(String val) {
         this(val, 10);
     }
 
     /**
      * Constructs a randomly generated BigInteger, uniformly distributed over
      * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
      * The uniformity of the distribution assumes that a fair source of random
      * bits is provided in {@code rnd}.  Note that this constructor always
      * constructs a non-negative BigInteger.
      *
      * @param  numBits maximum bitLength of the new BigInteger.
      * @param  rnd source of randomness to be used in computing the new
      *         BigInteger.
      * @throws IllegalArgumentException {@code numBits} is negative.
      * @see #bitLength()
      */
     public BigInteger(int numBits, Random rnd) {
         this(1, randomBits(numBits, rnd));
     }
 
     private static byte[] randomBits(int numBits, Random rnd) {
         if (numBits < 0)
             throw new IllegalArgumentException("numBits must be non-negative");
         int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
         byte[] randomBits = new byte[numBytes];
 
         // Generate random bytes and mask out any excess bits
         if (numBytes > 0) {
             rnd.nextBytes(randomBits);
             int excessBits = 8*numBytes - numBits;
             randomBits[0] &= (1 << (8-excessBits)) - 1;
         }
         return randomBits;
     }
 
     /**
      * Constructs a randomly generated positive BigInteger that is probably
      * prime, with the specified bitLength.
      *
      * <p>It is recommended that the {@link #probablePrime probablePrime}
      * method be used in preference to this constructor unless there
      * is a compelling need to specify a certainty.
      *
      * @param  bitLength bitLength of the returned BigInteger.
      * @param  certainty a measure of the uncertainty that the caller is
      *         willing to tolerate.  The probability that the new BigInteger
      *         represents a prime number will exceed
      *         (1 - 1/2<sup>{@code certainty}</sup>).  The execution time of
      *         this constructor is proportional to the value of this parameter.
      * @param  rnd source of random bits used to select candidates to be
      *         tested for primality.
      * @throws ArithmeticException {@code bitLength < 2}.
      * @see    #bitLength()
      */
     public BigInteger(int bitLength, int certainty, Random rnd) {
         BigInteger prime;
 
         if (bitLength < 2)
             throw new ArithmeticException("bitLength < 2");
         // The cutoff of 95 was chosen empirically for best performance
         prime = (bitLength < 95 ? smallPrime(bitLength, certainty, rnd)
                                 : largePrime(bitLength, certainty, rnd));
         signum = 1;
         mag = prime.mag;
     }
 
     // Minimum size in bits that the requested prime number has
     // before we use the large prime number generating algorithms
     private static final int SMALL_PRIME_THRESHOLD = 95;
 
     // Certainty required to meet the spec of probablePrime
     private static final int DEFAULT_PRIME_CERTAINTY = 100;
 
     /**
      * Returns a positive BigInteger that is probably prime, with the
      * specified bitLength. The probability that a BigInteger returned
      * by this method is composite does not exceed 2<sup>-100</sup>.
      *
      * @param  bitLength bitLength of the returned BigInteger.
      * @param  rnd source of random bits used to select candidates to be
      *         tested for primality.
      * @return a BigInteger of {@code bitLength} bits that is probably prime
      * @throws ArithmeticException {@code bitLength < 2}.
      * @see    #bitLength()
      * @since 1.4
      */
     public static BigInteger probablePrime(int bitLength, Random rnd) {
         if (bitLength < 2)
             throw new ArithmeticException("bitLength < 2");
 
         // The cutoff of 95 was chosen empirically for best performance
         return (bitLength < SMALL_PRIME_THRESHOLD ?
                 smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
                 largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
     }
 
     /**
      * Find a random number of the specified bitLength that is probably prime.
      * This method is used for smaller primes, its performance degrades on
      * larger bitlengths.
      *
      * This method assumes bitLength > 1.
      */
     private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
         int magLen = (bitLength + 31) >>> 5;
         int temp[] = new int[magLen];
         int highBit = 1 << ((bitLength+31) & 0x1f);  // High bit of high int
         int highMask = (highBit << 1) - 1;  // Bits to keep in high int
 
         while(true) {
             // Construct a candidate
             for (int i=0; i<magLen; i++)
                 temp[i] = rnd.nextInt();
             temp[0] = (temp[0] & highMask) | highBit;  // Ensure exact length
             if (bitLength > 2)
                 temp[magLen-1] |= 1;  // Make odd if bitlen > 2
 
             BigInteger p = new BigInteger(temp, 1);
 
             // Do cheap "pre-test" if applicable
             if (bitLength > 6) {
                 long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
                 if ((r%3==0)  || (r%5==0)  || (r%7==0)  || (r%11==0) ||
                     (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
                     (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
                     continue; // Candidate is composite; try another
             }
 
             // All candidates of bitLength 2 and 3 are prime by this point
             if (bitLength < 4)
                 return p;
 
             // Do expensive test if we survive pre-test (or it's inapplicable)
             if (p.primeToCertainty(certainty, rnd))
                 return p;
         }
     }
 
     private static final BigInteger SMALL_PRIME_PRODUCT
                        = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);
 
     /**
      * Find a random number of the specified bitLength that is probably prime.
      * This method is more appropriate for larger bitlengths since it uses
      * a sieve to eliminate most composites before using a more expensive
      * test.
      */
     private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
         BigInteger p;
         p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
         p.mag[p.mag.length-1] &= 0xfffffffe;
 
         // Use a sieve length likely to contain the next prime number
         int searchLen = (bitLength / 20) * 64;
         BitSieve searchSieve = new BitSieve(p, searchLen);
         BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
 
         while ((candidate == null) || (candidate.bitLength() != bitLength)) {
             p = p.add(BigInteger.valueOf(2*searchLen));
             if (p.bitLength() != bitLength)
                 p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
             p.mag[p.mag.length-1] &= 0xfffffffe;
             searchSieve = new BitSieve(p, searchLen);
             candidate = searchSieve.retrieve(p, certainty, rnd);
         }
         return candidate;
     }
 
    /**
     * Returns the first integer greater than this {@code BigInteger} that
     * is probably prime.  The probability that the number returned by this
     * method is composite does not exceed 2<sup>-100</sup>. This method will
     * never skip over a prime when searching: if it returns {@code p}, there
     * is no prime {@code q} such that {@code this < q < p}.
     *
     * @return the first integer greater than this {@code BigInteger} that
     *         is probably prime.
     * @throws ArithmeticException {@code this < 0}.
     * @since 1.5
     */
     public BigInteger nextProbablePrime() {
         if (this.signum < 0)
             throw new ArithmeticException("start < 0: " + this);
 
         // Handle trivial cases
         if ((this.signum == 0) || this.equals(ONE))
             return TWO;
 
         BigInteger result = this.add(ONE);
 
         // Fastpath for small numbers
         if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
 
             // Ensure an odd number
             if (!result.testBit(0))
                 result = result.add(ONE);
 
             while(true) {
                 // Do cheap "pre-test" if applicable
                 if (result.bitLength() > 6) {
                     long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
                     if ((r%3==0)  || (r%5==0)  || (r%7==0)  || (r%11==0) ||
                         (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
                         (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
                         result = result.add(TWO);
                         continue; // Candidate is composite; try another
                     }
                 }
 
                 // All candidates of bitLength 2 and 3 are prime by this point
                 if (result.bitLength() < 4)
                     return result;
 
                 // The expensive test
                 if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
                     return result;
 
                 result = result.add(TWO);
             }
         }
 
         // Start at previous even number
         if (result.testBit(0))
             result = result.subtract(ONE);
 
         // Looking for the next large prime
         int searchLen = (result.bitLength() / 20) * 64;
 
         while(true) {
            BitSieve searchSieve = new BitSieve(result, searchLen);
            BigInteger candidate = searchSieve.retrieve(result,
                                                  DEFAULT_PRIME_CERTAINTY, null);
            if (candidate != null)
                return candidate;
            result = result.add(BigInteger.valueOf(2 * searchLen));
         }
     }
 
     /**
      * Returns {@code true} if this BigInteger is probably prime,
      * {@code false} if it's definitely composite.
      *
      * This method assumes bitLength > 2.
      *
      * @param  certainty a measure of the uncertainty that the caller is
      *         willing to tolerate: if the call returns {@code true}
      *         the probability that this BigInteger is prime exceeds
      *         {@code (1 - 1/2<sup>certainty</sup>)}.  The execution time of
      *         this method is proportional to the value of this parameter.
      * @return {@code true} if this BigInteger is probably prime,
      *         {@code false} if it's definitely composite.
      */
     boolean primeToCertainty(int certainty, Random random) {
         int rounds = 0;
         int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
 
         // The relationship between the certainty and the number of rounds
         // we perform is given in the draft standard ANSI X9.80, "PRIME
         // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
         int sizeInBits = this.bitLength();
         if (sizeInBits < 100) {
             rounds = 50;
             rounds = n < rounds ? n : rounds;
             return passesMillerRabin(rounds, random);
         }
 
         if (sizeInBits < 256) {
             rounds = 27;
         } else if (sizeInBits < 512) {
             rounds = 15;
         } else if (sizeInBits < 768) {
             rounds = 8;
         } else if (sizeInBits < 1024) {
             rounds = 4;
         } else {
             rounds = 2;
         }
         rounds = n < rounds ? n : rounds;
 
         return passesMillerRabin(rounds, random) && passesLucasLehmer();
     }
 
     /**
      * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
      *
      * The following assumptions are made:
      * This BigInteger is a positive, odd number.
      */
     private boolean passesLucasLehmer() {
         BigInteger thisPlusOne = this.add(ONE);
 
         // Step 1
         int d = 5;
         while (jacobiSymbol(d, this) != -1) {
             // 5, -7, 9, -11, ...
             d = (d<0) ? Math.abs(d)+2 : -(d+2);
         }
 
         // Step 2
         BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
 
         // Step 3
         return u.mod(this).equals(ZERO);
     }
 
     /**
      * Computes Jacobi(p,n).
      * Assumes n positive, odd, n>=3.
      */
     private static int jacobiSymbol(int p, BigInteger n) {
         if (p == 0)
             return 0;
 
         // Algorithm and comments adapted from Colin Plumb's C library.
         int j = 1;
         int u = n.mag[n.mag.length-1];
 
         // Make p positive
         if (p < 0) {
             p = -p;
             int n8 = u & 7;
             if ((n8 == 3) || (n8 == 7))
                 j = -j; // 3 (011) or 7 (111) mod 8
         }
 
         // Get rid of factors of 2 in p
         while ((p & 3) == 0)
             p >>= 2;
         if ((p & 1) == 0) {
             p >>= 1;
             if (((u ^ (u>>1)) & 2) != 0)
                 j = -j; // 3 (011) or 5 (101) mod 8
         }
         if (p == 1)
             return j;
         // Then, apply quadratic reciprocity
         if ((p & u & 2) != 0)   // p = u = 3 (mod 4)?
             j = -j;
         // And reduce u mod p
         u = n.mod(BigInteger.valueOf(p)).intValue();
 
         // Now compute Jacobi(u,p), u < p
         while (u != 0) {
             while ((u & 3) == 0)
                 u >>= 2;
             if ((u & 1) == 0) {
                 u >>= 1;
                 if (((p ^ (p>>1)) & 2) != 0)
                     j = -j;     // 3 (011) or 5 (101) mod 8
             }
             if (u == 1)
                 return j;
             // Now both u and p are odd, so use quadratic reciprocity
             assert (u < p);
             int t = u; u = p; p = t;
             if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
                 j = -j;
             // Now u >= p, so it can be reduced
             u %= p;
         }
         return 0;
     }
 
     private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
         BigInteger d = BigInteger.valueOf(z);
         BigInteger u = ONE; BigInteger u2;
         BigInteger v = ONE; BigInteger v2;
 
         for (int i=k.bitLength()-2; i>=0; i--) {
             u2 = u.multiply(v).mod(n);
 
             v2 = v.square().add(d.multiply(u.square())).mod(n);
             if (v2.testBit(0))
                 v2 = v2.subtract(n);
 
             v2 = v2.shiftRight(1);
 
             u = u2; v = v2;
             if (k.testBit(i)) {
                 u2 = u.add(v).mod(n);
                 if (u2.testBit(0))
                     u2 = u2.subtract(n);
 
                 u2 = u2.shiftRight(1);
                 v2 = v.add(d.multiply(u)).mod(n);
                 if (v2.testBit(0))
                     v2 = v2.subtract(n);
                 v2 = v2.shiftRight(1);
 
                 u = u2; v = v2;
             }
         }
         return u;
     }
 
     private static volatile Random staticRandom;
 
     private static Random getSecureRandom() {
         if (staticRandom == null) {
             staticRandom = new java.security.SecureRandom();
         }
         return staticRandom;
     }
 
     /**
      * Returns true iff this BigInteger passes the specified number of
      * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
      * 186-2).
      *
      * The following assumptions are made:
      * This BigInteger is a positive, odd number greater than 2.
      * iterations<=50.
      */
     private boolean passesMillerRabin(int iterations, Random rnd) {
         // Find a and m such that m is odd and this == 1 + 2**a * m
         BigInteger thisMinusOne = this.subtract(ONE);
         BigInteger m = thisMinusOne;
         int a = m.getLowestSetBit();
         m = m.shiftRight(a);
 
         // Do the tests
         if (rnd == null) {
             rnd = getSecureRandom();
         }
         for (int i=0; i<iterations; i++) {
             // Generate a uniform random on (1, this)
             BigInteger b;
             do {
                 b = new BigInteger(this.bitLength(), rnd);
             } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);
 
             int j = 0;
             BigInteger z = b.modPow(m, this);
             while(!((j==0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
                 if (j>0 && z.equals(ONE) || ++j==a)
                     return false;
                 z = z.modPow(TWO, this);
             }
         }
         return true;
     }
 
     /**
      * This internal constructor differs from its public cousin
      * with the arguments reversed in two ways: it assumes that its
      * arguments are correct, and it doesn't copy the magnitude array.
      */
     BigInteger(int[] magnitude, int signum) {
         this.signum = (magnitude.length==0 ? 0 : signum);
         this.mag = magnitude;
     }
 
     /**
      * This private constructor is for internal use and assumes that its
      * arguments are correct.
      */
     private BigInteger(byte[] magnitude, int signum) {
         this.signum = (magnitude.length==0 ? 0 : signum);
         this.mag = stripLeadingZeroBytes(magnitude);
     }
 
     //Static Factory Methods
 
     /**
      * Returns a BigInteger whose value is equal to that of the
      * specified {@code long}.  This "static factory method" is
      * provided in preference to a ({@code long}) constructor
      * because it allows for reuse of frequently used BigIntegers.
      *
      * @param  val value of the BigInteger to return.
      * @return a BigInteger with the specified value.
      */
     public static BigInteger valueOf(long val) {
         // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
         if (val == 0)
             return ZERO;
         if (val > 0 && val <= MAX_CONSTANT)
             return posConst[(int) val];
         else if (val < 0 && val >= -MAX_CONSTANT)
             return negConst[(int) -val];
 
         return new BigInteger(val);
     }
 
     /**
      * Constructs a BigInteger with the specified value, which may not be zero.
      */
     private BigInteger(long val) {
         if (val < 0) {
             val = -val;
             signum = -1;
         } else {
             signum = 1;
         }
 
         int highWord = (int)(val >>> 32);
         if (highWord==0) {
             mag = new int[1];
             mag[0] = (int)val;
         } else {
             mag = new int[2];
             mag[0] = highWord;
             mag[1] = (int)val;
         }
     }
 
     /**
      * Returns a BigInteger with the given two's complement representation.
      * Assumes that the input array will not be modified (the returned
      * BigInteger will reference the input array if feasible).
      */
     private static BigInteger valueOf(int val[]) {
         return (val[0]>0 ? new BigInteger(val, 1) : new BigInteger(val));
     }
 
     // Constants
 
     /**
      * Initialize static constant array when class is loaded.
      */
     private final static int MAX_CONSTANT = 16;
     private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
     private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
     static {
         for (int i = 1; i <= MAX_CONSTANT; i++) {
             int[] magnitude = new int[1];
             magnitude[0] = i;
             posConst[i] = new BigInteger(magnitude,  1);
             negConst[i] = new BigInteger(magnitude, -1);
         }
     }
 
     /**
      * The BigInteger constant zero.
      *
      * @since   1.2
      */
     public static final BigInteger ZERO = new BigInteger(new int[0], 0);
 
     /**
      * The BigInteger constant one.
      *
      * @since   1.2
      */
     public static final BigInteger ONE = valueOf(1);
 
     /**
      * The BigInteger constant two.  (Not exported.)
      */
     private static final BigInteger TWO = valueOf(2);
 
     /**
      * The BigInteger constant ten.
      *
      * @since   1.5
      */
     public static final BigInteger TEN = valueOf(10);
 
     // Arithmetic Operations
 
     /**
      * Returns a BigInteger whose value is {@code (this + val)}.
      *
      * @param  val value to be added to this BigInteger.
      * @return {@code this + val}
      */
     public BigInteger add(BigInteger val) {
         if (val.signum == 0)
             return this;
         if (signum == 0)
             return val;
         if (val.signum == signum)
             return new BigInteger(add(mag, val.mag), signum);
 
         int cmp = compareMagnitude(val);
         if (cmp == 0)
             return ZERO;
         int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
                            : subtract(val.mag, mag));
         resultMag = trustedStripLeadingZeroInts(resultMag);
 
         return new BigInteger(resultMag, cmp == signum ? 1 : -1);
     }
 
     /**
      * Adds the contents of the int arrays x and y. This method allocates
      * a new int array to hold the answer and returns a reference to that
      * array.
      */
     private static int[] add(int[] x, int[] y) {
         // If x is shorter, swap the two arrays
         if (x.length < y.length) {
             int[] tmp = x;
             x = y;
             y = tmp;
         }
 
         int xIndex = x.length;
         int yIndex = y.length;
         int result[] = new int[xIndex];
         long sum = 0;
 
         // Add common parts of both numbers
         while(yIndex > 0) {
             sum = (x[--xIndex] & LONG_MASK) +
                   (y[--yIndex] & LONG_MASK) + (sum >>> 32);
             result[xIndex] = (int)sum;
         }
 
         // Copy remainder of longer number while carry propagation is required
         boolean carry = (sum >>> 32 != 0);
         while (xIndex > 0 && carry)
             carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
 
         // Copy remainder of longer number
         while (xIndex > 0)
             result[--xIndex] = x[xIndex];
 
         // Grow result if necessary
         if (carry) {
             int bigger[] = new int[result.length + 1];
             System.arraycopy(result, 0, bigger, 1, result.length);
             bigger[0] = 0x01;
             return bigger;
         }
         return result;
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this - val)}.
      *
      * @param  val value to be subtracted from this BigInteger.
      * @return {@code this - val}
      */
     public BigInteger subtract(BigInteger val) {
         if (val.signum == 0)
             return this;
         if (signum == 0)
             return val.negate();
         if (val.signum != signum)
             return new BigInteger(add(mag, val.mag), signum);
 
         int cmp = compareMagnitude(val);
         if (cmp == 0)
             return ZERO;
         int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
                            : subtract(val.mag, mag));
         resultMag = trustedStripLeadingZeroInts(resultMag);
         return new BigInteger(resultMag, cmp == signum ? 1 : -1);
     }
 
     /**
      * Subtracts the contents of the second int arrays (little) from the
      * first (big).  The first int array (big) must represent a larger number
      * than the second.  This method allocates the space necessary to hold the
      * answer.
      */
     private static int[] subtract(int[] big, int[] little) {
         int bigIndex = big.length;
         int result[] = new int[bigIndex];
         int littleIndex = little.length;
         long difference = 0;
 
         // Subtract common parts of both numbers
         while(littleIndex > 0) {
             difference = (big[--bigIndex] & LONG_MASK) -
                          (little[--littleIndex] & LONG_MASK) +
                          (difference >> 32);
             result[bigIndex] = (int)difference;
         }
 
         // Subtract remainder of longer number while borrow propagates
         boolean borrow = (difference >> 32 != 0);
         while (bigIndex > 0 && borrow)
             borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
 
         // Copy remainder of longer number
         while (bigIndex > 0)
             result[--bigIndex] = big[bigIndex];
 
         return result;
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this * val)}.
      *
      * @param  val value to be multiplied by this BigInteger.
      * @return {@code this * val}
      */
     public BigInteger multiply(BigInteger val) {
         if (val.signum == 0 || signum == 0)
             return ZERO;
 
         int[] result = multiplyToLen(mag, mag.length,
                                      val.mag, val.mag.length, null);
         result = trustedStripLeadingZeroInts(result);
         return new BigInteger(result, signum == val.signum ? 1 : -1);
     }
 
     /**
      * Package private methods used by BigDecimal code to multiply a BigInteger
      * with a long. Assumes v is not equal to INFLATED.
      */
     BigInteger multiply(long v) {
         if (v == 0 || signum == 0)
           return ZERO;
         if (v == BigDecimal.INFLATED)
             return multiply(BigInteger.valueOf(v));
         int rsign = (v > 0 ? signum : -signum);
         if (v < 0)
             v = -v;
         long dh = v >>> 32;      // higher order bits
         long dl = v & LONG_MASK; // lower order bits
 
         int xlen = mag.length;
         int[] value = mag;
         int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
         long carry = 0;
         int rstart = rmag.length - 1;
         for (int i = xlen - 1; i >= 0; i--) {
             long product = (value[i] & LONG_MASK) * dl + carry;
             rmag[rstart--] = (int)product;
             carry = product >>> 32;
         }
         rmag[rstart] = (int)carry;
         if (dh != 0L) {
             carry = 0;
             rstart = rmag.length - 2;
             for (int i = xlen - 1; i >= 0; i--) {
                 long product = (value[i] & LONG_MASK) * dh +
                     (rmag[rstart] & LONG_MASK) + carry;
                 rmag[rstart--] = (int)product;
                 carry = product >>> 32;
             }
             rmag[0] = (int)carry;
         }
         if (carry == 0L)
             rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
         return new BigInteger(rmag, rsign);
     }
 
     /**
      * Multiplies int arrays x and y to the specified lengths and places
      * the result into z. There will be no leading zeros in the resultant array.
      */
     private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
         int xstart = xlen - 1;
         int ystart = ylen - 1;
 
         if (z == null || z.length < (xlen+ ylen))
             z = new int[xlen+ylen];
 
         long carry = 0;
         for (int j=ystart, k=ystart+1+xstart; j>=0; j--, k--) {
             long product = (y[j] & LONG_MASK) *
                            (x[xstart] & LONG_MASK) + carry;
             z[k] = (int)product;
             carry = product >>> 32;
         }
         z[xstart] = (int)carry;
 
         for (int i = xstart-1; i >= 0; i--) {
             carry = 0;
             for (int j=ystart, k=ystart+1+i; j>=0; j--, k--) {
                 long product = (y[j] & LONG_MASK) *
                                (x[i] & LONG_MASK) +
                                (z[k] & LONG_MASK) + carry;
                 z[k] = (int)product;
                 carry = product >>> 32;
             }
             z[i] = (int)carry;
         }
         return z;
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
      *
      * @return {@code this<sup>2</sup>}
      */
     private BigInteger square() {
         if (signum == 0)
             return ZERO;
         int[] z = squareToLen(mag, mag.length, null);
         return new BigInteger(trustedStripLeadingZeroInts(z), 1);
     }
 
     /**
      * Squares the contents of the int array x. The result is placed into the
      * int array z.  The contents of x are not changed.
      */
     private static final int[] squareToLen(int[] x, int len, int[] z) {
         /*
          * The algorithm used here is adapted from Colin Plumb's C library.
          * Technique: Consider the partial products in the multiplication
          * of "abcde" by itself:
          *
          *               a  b  c  d  e
          *            *  a  b  c  d  e
          *          ==================
          *              ae be ce de ee
          *           ad bd cd dd de
          *        ac bc cc cd ce
          *     ab bb bc bd be
          *  aa ab ac ad ae
          *
          * Note that everything above the main diagonal:
          *              ae be ce de = (abcd) * e
          *           ad bd cd       = (abc) * d
          *        ac bc             = (ab) * c
          *     ab                   = (a) * b
          *
          * is a copy of everything below the main diagonal:
          *                       de
          *                 cd ce
          *           bc bd be
          *     ab ac ad ae
          *
          * Thus, the sum is 2 * (off the diagonal) + diagonal.
          *
          * This is accumulated beginning with the diagonal (which
          * consist of the squares of the digits of the input), which is then
          * divided by two, the off-diagonal added, and multiplied by two
          * again.  The low bit is simply a copy of the low bit of the
          * input, so it doesn't need special care.
          */
         int zlen = len << 1;
         if (z == null || z.length < zlen)
             z = new int[zlen];
 
         // Store the squares, right shifted one bit (i.e., divided by 2)
         int lastProductLowWord = 0;
         for (int j=0, i=0; j<len; j++) {
             long piece = (x[j] & LONG_MASK);
             long product = piece * piece;
             z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
             z[i++] = (int)(product >>> 1);
             lastProductLowWord = (int)product;
         }
 
         // Add in off-diagonal sums
         for (int i=len, offset=1; i>0; i--, offset+=2) {
             int t = x[i-1];
             t = mulAdd(z, x, offset, i-1, t);
             addOne(z, offset-1, i, t);
         }
 
         // Shift back up and set low bit
         primitiveLeftShift(z, zlen, 1);
         z[zlen-1] |= x[len-1] & 1;
 
         return z;
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this / val)}.
      *
      * @param  val value by which this BigInteger is to be divided.
      * @return {@code this / val}
      * @throws ArithmeticException if {@code val} is zero.
      */
     public BigInteger divide(BigInteger val) {
         MutableBigInteger q = new MutableBigInteger(),
                           a = new MutableBigInteger(this.mag),
                           b = new MutableBigInteger(val.mag);
 
         a.divide(b, q);
         return q.toBigInteger(this.signum == val.signum ? 1 : -1);
     }
 
     /**
      * Returns an array of two BigIntegers containing {@code (this / val)}
      * followed by {@code (this % val)}.
      *
      * @param  val value by which this BigInteger is to be divided, and the
      *         remainder computed.
      * @return an array of two BigIntegers: the quotient {@code (this / val)}
      *         is the initial element, and the remainder {@code (this % val)}
      *         is the final element.
      * @throws ArithmeticException if {@code val} is zero.
      */
     public BigInteger[] divideAndRemainder(BigInteger val) {
         BigInteger[] result = new BigInteger[2];
         MutableBigInteger q = new MutableBigInteger(),
                           a = new MutableBigInteger(this.mag),
                           b = new MutableBigInteger(val.mag);
         MutableBigInteger r = a.divide(b, q);
         result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
         result[1] = r.toBigInteger(this.signum);
         return result;
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this % val)}.
      *
      * @param  val value by which this BigInteger is to be divided, and the
      *         remainder computed.
      * @return {@code this % val}
      * @throws ArithmeticException if {@code val} is zero.
      */
     public BigInteger remainder(BigInteger val) {
         MutableBigInteger q = new MutableBigInteger(),
                           a = new MutableBigInteger(this.mag),
                           b = new MutableBigInteger(val.mag);
 
         return a.divide(b, q).toBigInteger(this.signum);
     }
 
     /**
      * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
      * Note that {@code exponent} is an integer rather than a BigInteger.
      *
      * @param  exponent exponent to which this BigInteger is to be raised.
      * @return <tt>this<sup>exponent</sup></tt>
      * @throws ArithmeticException {@code exponent} is negative.  (This would
      *         cause the operation to yield a non-integer value.)
      */
     public BigInteger pow(int exponent) {
         if (exponent < 0)
             throw new ArithmeticException("Negative exponent");
         if (signum==0)
             return (exponent==0 ? ONE : this);
 
         // Perform exponentiation using repeated squaring trick
         int newSign = (signum<0 && (exponent&1)==1 ? -1 : 1);
         int[] baseToPow2 = this.mag;
         int[] result = {1};
 
         while (exponent != 0) {
             if ((exponent & 1)==1) {
                 result = multiplyToLen(result, result.length,
                                        baseToPow2, baseToPow2.length, null);
                 result = trustedStripLeadingZeroInts(result);
             }
             if ((exponent >>>= 1) != 0) {
                 baseToPow2 = squareToLen(baseToPow2, baseToPow2.length, null);
                 baseToPow2 = trustedStripLeadingZeroInts(baseToPow2);
             }
         }
         return new BigInteger(result, newSign);
     }
 
     /**
      * Returns a BigInteger whose value is the greatest common divisor of
      * {@code abs(this)} and {@code abs(val)}.  Returns 0 if
      * {@code this==0 && val==0}.
      *
      * @param  val value with which the GCD is to be computed.
      * @return {@code GCD(abs(this), abs(val))}
      */
     public BigInteger gcd(BigInteger val) {
         if (val.signum == 0)
             return this.abs();
         else if (this.signum == 0)
             return val.abs();
 
         MutableBigInteger a = new MutableBigInteger(this);
         MutableBigInteger b = new MutableBigInteger(val);
 
         MutableBigInteger result = a.hybridGCD(b);
 
         return result.toBigInteger(1);
     }
 
     /**
      * Package private method to return bit length for an integer.
      */
     static int bitLengthForInt(int n) {
         return 32 - Integer.numberOfLeadingZeros(n);
     }
 
     /**
      * Left shift int array a up to len by n bits. Returns the array that
      * results from the shift since space may have to be reallocated.
      */
     private static int[] leftShift(int[] a, int len, int n) {
         int nInts = n >>> 5;
         int nBits = n&0x1F;
         int bitsInHighWord = bitLengthForInt(a[0]);
 
         // If shift can be done without recopy, do so
         if (n <= (32-bitsInHighWord)) {
             primitiveLeftShift(a, len, nBits);
             return a;
         } else { // Array must be resized
             if (nBits <= (32-bitsInHighWord)) {
                 int result[] = new int[nInts+len];
                 for (int i=0; i<len; i++)
                     result[i] = a[i];
                 primitiveLeftShift(result, result.length, nBits);
                 return result;
             } else {
                 int result[] = new int[nInts+len+1];
                 for (int i=0; i<len; i++)
                     result[i] = a[i];
                 primitiveRightShift(result, result.length, 32 - nBits);
                 return result;
             }
         }
     }
 
     // shifts a up to len right n bits assumes no leading zeros, 0<n<32
     static void primitiveRightShift(int[] a, int len, int n) {
         int n2 = 32 - n;
         for (int i=len-1, c=a[i]; i>0; i--) {
             int b = c;
             c = a[i-1];
             a[i] = (c << n2) | (b >>> n);
         }
         a[0] >>>= n;
     }
 
     // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
     static void primitiveLeftShift(int[] a, int len, int n) {
         if (len == 0 || n == 0)
             return;
 
         int n2 = 32 - n;
         for (int i=0, c=a[i], m=i+len-1; i<m; i++) {
             int b = c;
             c = a[i+1];
             a[i] = (b << n) | (c >>> n2);
         }
         a[len-1] <<= n;
     }
 
     /**
      * Calculate bitlength of contents of the first len elements an int array,
      * assuming there are no leading zero ints.
      */
     private static int bitLength(int[] val, int len) {
         if (len == 0)
             return 0;
         return ((len - 1) << 5) + bitLengthForInt(val[0]);
     }
 
     /**
      * Returns a BigInteger whose value is the absolute value of this
      * BigInteger.
      *
      * @return {@code abs(this)}
      */
     public BigInteger abs() {
         return (signum >= 0 ? this : this.negate());
     }
 
     /**
      * Returns a BigInteger whose value is {@code (-this)}.
      *
      * @return {@code -this}
      */
     public BigInteger negate() {
         return new BigInteger(this.mag, -this.signum);
     }
 
     /**
      * Returns the signum function of this BigInteger.
      *
      * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
      *         positive.
      */
     public int signum() {
         return this.signum;
     }
 
     // Modular Arithmetic Operations
 
     /**
      * Returns a BigInteger whose value is {@code (this mod m}).  This method
      * differs from {@code remainder} in that it always returns a
      * <i>non-negative</i> BigInteger.
      *
      * @param  m the modulus.
      * @return {@code this mod m}
      * @throws ArithmeticException {@code m} &le; 0
      * @see    #remainder
      */
     public BigInteger mod(BigInteger m) {
         if (m.signum <= 0)
             throw new ArithmeticException("BigInteger: modulus not positive");
 
         BigInteger result = this.remainder(m);
         return (result.signum >= 0 ? result : result.add(m));
     }
 
     /**
      * Returns a BigInteger whose value is
      * <tt>(this<sup>exponent</sup> mod m)</tt>.  (Unlike {@code pow}, this
      * method permits negative exponents.)
      *
      * @param  exponent the exponent.
      * @param  m the modulus.
      * @return <tt>this<sup>exponent</sup> mod m</tt>
      * @throws ArithmeticException {@code m} &le; 0 or the exponent is
      *         negative and this BigInteger is not <i>relatively
      *         prime</i> to {@code m}.
      * @see    #modInverse
      */
     public BigInteger modPow(BigInteger exponent, BigInteger m) {
         if (m.signum <= 0)
             throw new ArithmeticException("BigInteger: modulus not positive");
 
         // Trivial cases
         if (exponent.signum == 0)
             return (m.equals(ONE) ? ZERO : ONE);
 
         if (this.equals(ONE))
             return (m.equals(ONE) ? ZERO : ONE);
 
         if (this.equals(ZERO) && exponent.signum >= 0)
             return ZERO;
 
         if (this.equals(negConst[1]) && (!exponent.testBit(0)))
             return (m.equals(ONE) ? ZERO : ONE);
 
         boolean invertResult;
         if ((invertResult = (exponent.signum < 0)))
             exponent = exponent.negate();
 
         BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
                            ? this.mod(m) : this);
         BigInteger result;
         if (m.testBit(0)) { // odd modulus
             result = base.oddModPow(exponent, m);
         } else {
             /*
              * Even modulus.  Tear it into an "odd part" (m1) and power of two
              * (m2), exponentiate mod m1, manually exponentiate mod m2, and
              * use Chinese Remainder Theorem to combine results.
              */
 
             // Tear m apart into odd part (m1) and power of 2 (m2)
             int p = m.getLowestSetBit();   // Max pow of 2 that divides m
 
             BigInteger m1 = m.shiftRight(p);  // m/2**p
             BigInteger m2 = ONE.shiftLeft(p); // 2**p
 
             // Calculate new base from m1
             BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
                                 ? this.mod(m1) : this);
 
             // Caculate (base ** exponent) mod m1.
             BigInteger a1 = (m1.equals(ONE) ? ZERO :
                              base2.oddModPow(exponent, m1));
 
             // Calculate (this ** exponent) mod m2
             BigInteger a2 = base.modPow2(exponent, p);
 
             // Combine results using Chinese Remainder Theorem
             BigInteger y1 = m2.modInverse(m1);
             BigInteger y2 = m1.modInverse(m2);
 
             result = a1.multiply(m2).multiply(y1).add
                      (a2.multiply(m1).multiply(y2)).mod(m);
         }
 
         return (invertResult ? result.modInverse(m) : result);
     }
 
     static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
                                                 Integer.MAX_VALUE}; // Sentinel
 
     /**
      * Returns a BigInteger whose value is x to the power of y mod z.
      * Assumes: z is odd && x < z.
      */
     private BigInteger oddModPow(BigInteger y, BigInteger z) {
     /*
      * The algorithm is adapted from Colin Plumb's C library.
      *
      * The window algorithm:
      * The idea is to keep a running product of b1 = n^(high-order bits of exp)
      * and then keep appending exponent bits to it.  The following patterns
      * apply to a 3-bit window (k = 3):
      * To append   0: square
      * To append   1: square, multiply by n^1
      * To append  10: square, multiply by n^1, square
      * To append  11: square, square, multiply by n^3
      * To append 100: square, multiply by n^1, square, square
      * To append 101: square, square, square, multiply by n^5
      * To append 110: square, square, multiply by n^3, square
      * To append 111: square, square, square, multiply by n^7
      *
      * Since each pattern involves only one multiply, the longer the pattern
      * the better, except that a 0 (no multiplies) can be appended directly.
      * We precompute a table of odd powers of n, up to 2^k, and can then
      * multiply k bits of exponent at a time.  Actually, assuming random
      * exponents, there is on average one zero bit between needs to
      * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
      * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
      * you have to do one multiply per k+1 bits of exponent.
      *
      * The loop walks down the exponent, squaring the result buffer as
      * it goes.  There is a wbits+1 bit lookahead buffer, buf, that is
      * filled with the upcoming exponent bits.  (What is read after the
      * end of the exponent is unimportant, but it is filled with zero here.)
      * When the most-significant bit of this buffer becomes set, i.e.
      * (buf & tblmask) != 0, we have to decide what pattern to multiply
      * by, and when to do it.  We decide, remember to do it in future
      * after a suitable number of squarings have passed (e.g. a pattern
      * of "100" in the buffer requires that we multiply by n^1 immediately;
      * a pattern of "110" calls for multiplying by n^3 after one more
      * squaring), clear the buffer, and continue.
      *
      * When we start, there is one more optimization: the result buffer
      * is implcitly one, so squaring it or multiplying by it can be
      * optimized away.  Further, if we start with a pattern like "100"
      * in the lookahead window, rather than placing n into the buffer
      * and then starting to square it, we have already computed n^2
      * to compute the odd-powers table, so we can place that into
      * the buffer and save a squaring.
      *
      * This means that if you have a k-bit window, to compute n^z,
      * where z is the high k bits of the exponent, 1/2 of the time
      * it requires no squarings.  1/4 of the time, it requires 1
      * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
      * And the remaining 1/2^(k-1) of the time, the top k bits are a
      * 1 followed by k-1 0 bits, so it again only requires k-2
      * squarings, not k-1.  The average of these is 1.  Add that
      * to the one squaring we have to do to compute the table,
      * and you'll see that a k-bit window saves k-2 squarings
      * as well as reducing the multiplies.  (It actually doesn't
      * hurt in the case k = 1, either.)
      */
         // Special case for exponent of one
         if (y.equals(ONE))
             return this;
 
         // Special case for base of zero
         if (signum==0)
             return ZERO;
 
         int[] base = mag.clone();
         int[] exp = y.mag;
         int[] mod = z.mag;
         int modLen = mod.length;
 
         // Select an appropriate window size
         int wbits = 0;
         int ebits = bitLength(exp, exp.length);
         // if exponent is 65537 (0x10001), use minimum window size
         if ((ebits != 17) || (exp[0] != 65537)) {
             while (ebits > bnExpModThreshTable[wbits]) {
                 wbits++;
             }
         }
 
         // Calculate appropriate table size
         int tblmask = 1 << wbits;
 
         // Allocate table for precomputed odd powers of base in Montgomery form
         int[][] table = new int[tblmask][];
         for (int i=0; i<tblmask; i++)
             table[i] = new int[modLen];
 
         // Compute the modular inverse
         int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]);
 
         // Convert base to Montgomery form
         int[] a = leftShift(base, base.length, modLen << 5);
 
         MutableBigInteger q = new MutableBigInteger(),
                           a2 = new MutableBigInteger(a),
                           b2 = new MutableBigInteger(mod);
 
         MutableBigInteger r= a2.divide(b2, q);
         table[0] = r.toIntArray();
 
         // Pad table[0] with leading zeros so its length is at least modLen
         if (table[0].length < modLen) {
            int offset = modLen - table[0].length;
            int[] t2 = new int[modLen];
            for (int i=0; i<table[0].length; i++)
                t2[i+offset] = table[0][i];
            table[0] = t2;
         }
 
         // Set b to the square of the base
         int[] b = squareToLen(table[0], modLen, null);
         b = montReduce(b, mod, modLen, inv);
 
         // Set t to high half of b
         int[] t = new int[modLen];
         for(int i=0; i<modLen; i++)
             t[i] = b[i];
 
         // Fill in the table with odd powers of the base
         for (int i=1; i<tblmask; i++) {
             int[] prod = multiplyToLen(t, modLen, table[i-1], modLen, null);
             table[i] = montReduce(prod, mod, modLen, inv);
         }
 
         // Pre load the window that slides over the exponent
         int bitpos = 1 << ((ebits-1) & (32-1));
 
         int buf = 0;
         int elen = exp.length;
         int eIndex = 0;
         for (int i = 0; i <= wbits; i++) {
             buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0);
             bitpos >>>= 1;
             if (bitpos == 0) {
                 eIndex++;
                 bitpos = 1 << (32-1);
                 elen--;
             }
         }
 
         int multpos = ebits;
 
         // The first iteration, which is hoisted out of the main loop
         ebits--;
         boolean isone = true;
 
         multpos = ebits - wbits;
         while ((buf & 1) == 0) {
             buf >>>= 1;
             multpos++;
         }
 
         int[] mult = table[buf >>> 1];
 
         buf = 0;
         if (multpos == ebits)
             isone = false;
 
         // The main loop
         while(true) {
             ebits--;
             // Advance the window
             buf <<= 1;
 
             if (elen != 0) {
                 buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
                 bitpos >>>= 1;
                 if (bitpos == 0) {
                     eIndex++;
                     bitpos = 1 << (32-1);
                     elen--;
                 }
             }
 
             // Examine the window for pending multiplies
             if ((buf & tblmask) != 0) {
                 multpos = ebits - wbits;
                 while ((buf & 1) == 0) {
                     buf >>>= 1;
                     multpos++;
                 }
                 mult = table[buf >>> 1];
                 buf = 0;
             }
 
             // Perform multiply
             if (ebits == multpos) {
                 if (isone) {
                     b = mult.clone();
                     isone = false;
                 } else {
                     t = b;
                     a = multiplyToLen(t, modLen, mult, modLen, a);
                     a = montReduce(a, mod, modLen, inv);
                     t = a; a = b; b = t;
                 }
             }
 
             // Check if done
             if (ebits == 0)
                 break;
 
             // Square the input
             if (!isone) {
                 t = b;
                 a = squareToLen(t, modLen, a);
                 a = montReduce(a, mod, modLen, inv);
                 t = a; a = b; b = t;
             }
         }
 
         // Convert result out of Montgomery form and return
         int[] t2 = new int[2*modLen];
         for(int i=0; i<modLen; i++)
             t2[i+modLen] = b[i];
 
         b = montReduce(t2, mod, modLen, inv);
 
         t2 = new int[modLen];
         for(int i=0; i<modLen; i++)
             t2[i] = b[i];
 
         return new BigInteger(1, t2);
     }
 
     /**
      * Montgomery reduce n, modulo mod.  This reduces modulo mod and divides
      * by 2^(32*mlen). Adapted from Colin Plumb's C library.
      */
     private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
         int c=0;
         int len = mlen;
         int offset=0;
 
         do {
             int nEnd = n[n.length-1-offset];
             int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
             c += addOne(n, offset, mlen, carry);
             offset++;
         } while(--len > 0);
 
         while(c>0)
             c += subN(n, mod, mlen);
 
         while (intArrayCmpToLen(n, mod, mlen) >= 0)
             subN(n, mod, mlen);
 
         return n;
     }
 
 
     /*
      * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
      * equal to, or greater than arg2 up to length len.
      */
     private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
         for (int i=0; i<len; i++) {
             long b1 = arg1[i] & LONG_MASK;
             long b2 = arg2[i] & LONG_MASK;
             if (b1 < b2)
                 return -1;
             if (b1 > b2)
                 return 1;
         }
         return 0;
     }
 
     /**
      * Subtracts two numbers of same length, returning borrow.
      */
     private static int subN(int[] a, int[] b, int len) {
         long sum = 0;
 
         while(--len >= 0) {
             sum = (a[len] & LONG_MASK) -
                  (b[len] & LONG_MASK) + (sum >> 32);
             a[len] = (int)sum;
         }
 
         return (int)(sum >> 32);
     }
 
     /**
      * Multiply an array by one word k and add to result, return the carry
      */
     static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
         long kLong = k & LONG_MASK;
         long carry = 0;
 
         offset = out.length-offset - 1;
         for (int j=len-1; j >= 0; j--) {
             long product = (in[j] & LONG_MASK) * kLong +
                            (out[offset] & LONG_MASK) + carry;
             out[offset--] = (int)product;
             carry = product >>> 32;
         }
         return (int)carry;
     }
 
     /**
      * Add one word to the number a mlen words into a. Return the resulting
      * carry.
      */
     static int addOne(int[] a, int offset, int mlen, int carry) {
         offset = a.length-1-mlen-offset;
         long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);
 
         a[offset] = (int)t;
         if ((t >>> 32) == 0)
             return 0;
         while (--mlen >= 0) {
             if (--offset < 0) { // Carry out of number
                 return 1;
             } else {
                 a[offset]++;
                 if (a[offset] != 0)
                     return 0;
             }
         }
         return 1;
     }
 
     /**
      * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
      */
     private BigInteger modPow2(BigInteger exponent, int p) {
         /*
          * Perform exponentiation using repeated squaring trick, chopping off
          * high order bits as indicated by modulus.
          */
         BigInteger result = valueOf(1);
         BigInteger baseToPow2 = this.mod2(p);
         int expOffset = 0;
 
         int limit = exponent.bitLength();
 
         if (this.testBit(0))
            limit = (p-1) < limit ? (p-1) : limit;
 
         while (expOffset < limit) {
             if (exponent.testBit(expOffset))
                 result = result.multiply(baseToPow2).mod2(p);
             expOffset++;
             if (expOffset < limit)
                 baseToPow2 = baseToPow2.square().mod2(p);
         }
 
         return result;
     }
 
     /**
      * Returns a BigInteger whose value is this mod(2**p).
      * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
      */
     private BigInteger mod2(int p) {
         if (bitLength() <= p)
             return this;
 
         // Copy remaining ints of mag
         int numInts = (p + 31) >>> 5;
         int[] mag = new int[numInts];
         for (int i=0; i<numInts; i++)
             mag[i] = this.mag[i + (this.mag.length - numInts)];
 
         // Mask out any excess bits
         int excessBits = (numInts << 5) - p;
         mag[0] &= (1L << (32-excessBits)) - 1;
 
         return (mag[0]==0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
      *
      * @param  m the modulus.
      * @return {@code this}<sup>-1</sup> {@code mod m}.
      * @throws ArithmeticException {@code  m} &le; 0, or this BigInteger
      *         has no multiplicative inverse mod m (that is, this BigInteger
      *         is not <i>relatively prime</i> to m).
      */
     public BigInteger modInverse(BigInteger m) {
         if (m.signum != 1)
             throw new ArithmeticException("BigInteger: modulus not positive");
 
         if (m.equals(ONE))
             return ZERO;
 
         // Calculate (this mod m)
         BigInteger modVal = this;
         if (signum < 0 || (this.compareMagnitude(m) >= 0))
             modVal = this.mod(m);
 
         if (modVal.equals(ONE))
             return ONE;
 
         MutableBigInteger a = new MutableBigInteger(modVal);
         MutableBigInteger b = new MutableBigInteger(m);
 
         MutableBigInteger result = a.mutableModInverse(b);
         return result.toBigInteger(1);
     }
 
     // Shift Operations
 
     /**
      * Returns a BigInteger whose value is {@code (this << n)}.
      * The shift distance, {@code n}, may be negative, in which case
      * this method performs a right shift.
      * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
      *
      * @param  n shift distance, in bits.
      * @return {@code this << n}
      * @throws ArithmeticException if the shift distance is {@code
      *         Integer.MIN_VALUE}.
      * @see #shiftRight
      */
     public BigInteger shiftLeft(int n) {
         if (signum == 0)
             return ZERO;
         if (n==0)
             return this;
         if (n<0) {
             if (n == Integer.MIN_VALUE) {
                 throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
             } else {
                 return shiftRight(-n);
             }
         }
 
         int nInts = n >>> 5;
         int nBits = n & 0x1f;
         int magLen = mag.length;
         int newMag[] = null;
 
         if (nBits == 0) {
             newMag = new int[magLen + nInts];
             for (int i=0; i<magLen; i++)
                 newMag[i] = mag[i];
         } else {
             int i = 0;
             int nBits2 = 32 - nBits;
             int highBits = mag[0] >>> nBits2;
             if (highBits != 0) {
                 newMag = new int[magLen + nInts + 1];
                 newMag[i++] = highBits;
             } else {
                 newMag = new int[magLen + nInts];
             }
             int j=0;
             while (j < magLen-1)
                 newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
             newMag[i] = mag[j] << nBits;
         }
 
         return new BigInteger(newMag, signum);
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this >> n)}.  Sign
      * extension is performed.  The shift distance, {@code n}, may be
      * negative, in which case this method performs a left shift.
      * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.)
      *
      * @param  n shift distance, in bits.
      * @return {@code this >> n}
      * @throws ArithmeticException if the shift distance is {@code
      *         Integer.MIN_VALUE}.
      * @see #shiftLeft
      */
     public BigInteger shiftRight(int n) {
         if (n==0)
             return this;
         if (n<0) {
             if (n == Integer.MIN_VALUE) {
                 throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
             } else {
                 return shiftLeft(-n);
             }
         }
 
         int nInts = n >>> 5;
         int nBits = n & 0x1f;
         int magLen = mag.length;
         int newMag[] = null;
 
         // Special case: entire contents shifted off the end
         if (nInts >= magLen)
             return (signum >= 0 ? ZERO : negConst[1]);
 
         if (nBits == 0) {
             int newMagLen = magLen - nInts;
             newMag = new int[newMagLen];
             for (int i=0; i<newMagLen; i++)
                 newMag[i] = mag[i];
         } else {
             int i = 0;
             int highBits = mag[0] >>> nBits;
             if (highBits != 0) {
                 newMag = new int[magLen - nInts];
                 newMag[i++] = highBits;
             } else {
                 newMag = new int[magLen - nInts -1];
             }
 
             int nBits2 = 32 - nBits;
             int j=0;
             while (j < magLen - nInts - 1)
                 newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
         }
 
         if (signum < 0) {
             // Find out whether any one-bits were shifted off the end.
             boolean onesLost = false;
             for (int i=magLen-1, j=magLen-nInts; i>=j && !onesLost; i--)
                 onesLost = (mag[i] != 0);
             if (!onesLost && nBits != 0)
                 onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);
 
             if (onesLost)
                 newMag = javaIncrement(newMag);
         }
 
         return new BigInteger(newMag, signum);
     }
 
     int[] javaIncrement(int[] val) {
         int lastSum = 0;
         for (int i=val.length-1;  i >= 0 && lastSum == 0; i--)
             lastSum = (val[i] += 1);
         if (lastSum == 0) {
             val = new int[val.length+1];
             val[0] = 1;
         }
         return val;
     }
 
     // Bitwise Operations
 
     /**
      * Returns a BigInteger whose value is {@code (this & val)}.  (This
      * method returns a negative BigInteger if and only if this and val are
      * both negative.)
      *
      * @param val value to be AND'ed with this BigInteger.
      * @return {@code this & val}
      */
     public BigInteger and(BigInteger val) {
         int[] result = new int[Math.max(intLength(), val.intLength())];
         for (int i=0; i<result.length; i++)
             result[i] = (getInt(result.length-i-1)
                          & val.getInt(result.length-i-1));
 
         return valueOf(result);
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this | val)}.  (This method
      * returns a negative BigInteger if and only if either this or val is
      * negative.)
      *
      * @param val value to be OR'ed with this BigInteger.
      * @return {@code this | val}
      */
     public BigInteger or(BigInteger val) {
         int[] result = new int[Math.max(intLength(), val.intLength())];
         for (int i=0; i<result.length; i++)
             result[i] = (getInt(result.length-i-1)
                          | val.getInt(result.length-i-1));
 
         return valueOf(result);
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this ^ val)}.  (This method
      * returns a negative BigInteger if and only if exactly one of this and
      * val are negative.)
      *
      * @param val value to be XOR'ed with this BigInteger.
      * @return {@code this ^ val}
      */
     public BigInteger xor(BigInteger val) {
         int[] result = new int[Math.max(intLength(), val.intLength())];
         for (int i=0; i<result.length; i++)
             result[i] = (getInt(result.length-i-1)
                          ^ val.getInt(result.length-i-1));
 
         return valueOf(result);
     }
 
     /**
      * Returns a BigInteger whose value is {@code (~this)}.  (This method
      * returns a negative value if and only if this BigInteger is
      * non-negative.)
      *
      * @return {@code ~this}
      */
     public BigInteger not() {
         int[] result = new int[intLength()];
         for (int i=0; i<result.length; i++)
             result[i] = ~getInt(result.length-i-1);
 
         return valueOf(result);
     }
 
     /**
      * Returns a BigInteger whose value is {@code (this & ~val)}.  This
      * method, which is equivalent to {@code and(val.not())}, is provided as
      * a convenience for masking operations.  (This method returns a negative
      * BigInteger if and only if {@code this} is negative and {@code val} is
      * positive.)
      *
      * @param val value to be complemented and AND'ed with this BigInteger.
      * @return {@code this & ~val}
      */
     public BigInteger andNot(BigInteger val) {
         int[] result = new int[Math.max(intLength(), val.intLength())];
         for (int i=0; i<result.length; i++)
             result[i] = (getInt(result.length-i-1)
                          & ~val.getInt(result.length-i-1));
 
         return valueOf(result);
     }
 
 
     // Single Bit Operations
 
     /**
      * Returns {@code true} if and only if the designated bit is set.
      * (Computes {@code ((this & (1<<n)) != 0)}.)
      *
      * @param  n index of bit to test.
      * @return {@code true} if and only if the designated bit is set.
      * @throws ArithmeticException {@code n} is negative.
      */
     public boolean testBit(int n) {
         if (n<0)
             throw new ArithmeticException("Negative bit address");
 
         return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
     }
 
     /**
      * Returns a BigInteger whose value is equivalent to this BigInteger
      * with the designated bit set.  (Computes {@code (this | (1<<n))}.)
      *
      * @param  n index of bit to set.
      * @return {@code this | (1<<n)}
      * @throws ArithmeticException {@code n} is negative.
      */
     public BigInteger setBit(int n) {
         if (n<0)
             throw new ArithmeticException("Negative bit address");
 
         int intNum = n >>> 5;
         int[] result = new int[Math.max(intLength(), intNum+2)];
 
         for (int i=0; i<result.length; i++)
             result[result.length-i-1] = getInt(i);
 
         result[result.length-intNum-1] |= (1 << (n & 31));
 
         return valueOf(result);
     }
 
     /**
      * Returns a BigInteger whose value is equivalent to this BigInteger
      * with the designated bit cleared.
      * (Computes {@code (this & ~(1<<n))}.)
      *
      * @param  n index of bit to clear.
      * @return {@code this & ~(1<<n)}
      * @throws ArithmeticException {@code n} is negative.
      */
     public BigInteger clearBit(int n) {
         if (n<0)
             throw new ArithmeticException("Negative bit address");
 
         int intNum = n >>> 5;
         int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];
 
         for (int i=0; i<result.length; i++)
             result[result.length-i-1] = getInt(i);
 
         result[result.length-intNum-1] &= ~(1 << (n & 31));
 
         return valueOf(result);
     }
 
     /**
      * Returns a BigInteger whose value is equivalent to this BigInteger
      * with the designated bit flipped.
      * (Computes {@code (this ^ (1<<n))}.)
      *
      * @param  n index of bit to flip.
      * @return {@code this ^ (1<<n)}
      * @throws ArithmeticException {@code n} is negative.
      */
     public BigInteger flipBit(int n) {
         if (n<0)
             throw new ArithmeticException("Negative bit address");
 
         int intNum = n >>> 5;
         int[] result = new int[Math.max(intLength(), intNum+2)];
 
         for (int i=0; i<result.length; i++)
             result[result.length-i-1] = getInt(i);
 
         result[result.length-intNum-1] ^= (1 << (n & 31));
 
         return valueOf(result);
     }
 
     /**
      * Returns the index of the rightmost (lowest-order) one bit in this
      * BigInteger (the number of zero bits to the right of the rightmost
      * one bit).  Returns -1 if this BigInteger contains no one bits.
      * (Computes {@code (this==0? -1 : log2(this & -this))}.)
      *
      * @return index of the rightmost one bit in this BigInteger.
      */
     public int getLowestSetBit() {
         @SuppressWarnings("deprecation") int lsb = lowestSetBit - 2;
         if (lsb == -2) {  // lowestSetBit not initialized yet
             lsb = 0;
             if (signum == 0) {
                 lsb -= 1;
             } else {
                 // Search for lowest order nonzero int
                 int i,b;
                 for (i=0; (b = getInt(i))==0; i++)
                     ;
                 lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
             }
             lowestSetBit = lsb + 2;
         }
         return lsb;
     }
 
 
     // Miscellaneous Bit Operations
 
     /**
      * Returns the number of bits in the minimal two's-complement
      * representation of this BigInteger, <i>excluding</i> a sign bit.
      * For positive BigIntegers, this is equivalent to the number of bits in
      * the ordinary binary representation.  (Computes
      * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
      *
      * @return number of bits in the minimal two's-complement
      *         representation of this BigInteger, <i>excluding</i> a sign bit.
      */
     public int bitLength() {
         @SuppressWarnings("deprecation") int n = bitLength - 1;
         if (n == -1) { // bitLength not initialized yet
             int[] m = mag;
             int len = m.length;
             if (len == 0) {
                 n = 0; // offset by one to initialize
             }  else {
                 // Calculate the bit length of the magnitude
                 int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
                  if (signum < 0) {
                      // Check if magnitude is a power of two
                      boolean pow2 = (Integer.bitCount(mag[0]) == 1);
                      for(int i=1; i< len && pow2; i++)
                          pow2 = (mag[i] == 0);
 
                      n = (pow2 ? magBitLength -1 : magBitLength);
                  } else {
                      n = magBitLength;
                  }
             }
             bitLength = n + 1;
         }
         return n;
     }
 
     /**
      * Returns the number of bits in the two's complement representation
      * of this BigInteger that differ from its sign bit.  This method is
      * useful when implementing bit-vector style sets atop BigIntegers.
      *
      * @return number of bits in the two's complement representation
      *         of this BigInteger that differ from its sign bit.
      */
     public int bitCount() {
         @SuppressWarnings("deprecation") int bc = bitCount - 1;
         if (bc == -1) {  // bitCount not initialized yet
             bc = 0;      // offset by one to initialize
             // Count the bits in the magnitude
             for (int i=0; i<mag.length; i++)
                 bc += Integer.bitCount(mag[i]);
             if (signum < 0) {
                 // Count the trailing zeros in the magnitude
                 int magTrailingZeroCount = 0, j;
                 for (j=mag.length-1; mag[j]==0; j--)
                     magTrailingZeroCount += 32;
                 magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
                 bc += magTrailingZeroCount - 1;
             }
             bitCount = bc + 1;
         }
         return bc;
     }
 
     // Primality Testing
 
     /**
      * Returns {@code true} if this BigInteger is probably prime,
      * {@code false} if it's definitely composite.  If
      * {@code certainty} is &le; 0, {@code true} is
      * returned.
      *
      * @param  certainty a measure of the uncertainty that the caller is
      *         willing to tolerate: if the call returns {@code true}
      *         the probability that this BigInteger is prime exceeds
      *         (1 - 1/2<sup>{@code certainty}</sup>).  The execution time of
      *         this method is proportional to the value of this parameter.
      * @return {@code true} if this BigInteger is probably prime,
      *         {@code false} if it's definitely composite.
      */
     public boolean isProbablePrime(int certainty) {
         if (certainty <= 0)
             return true;
         BigInteger w = this.abs();
         if (w.equals(TWO))
             return true;
         if (!w.testBit(0) || w.equals(ONE))
             return false;
 
         return w.primeToCertainty(certainty, null);
     }
 
     // Comparison Operations
 
     /**
      * Compares this BigInteger with the specified BigInteger.  This
      * method is provided in preference to individual methods for each
      * of the six boolean comparison operators ({@literal <}, ==,
      * {@literal >}, {@literal >=}, !=, {@literal <=}).  The suggested
      * idiom for performing these comparisons is: {@code
      * (x.compareTo(y)} &lt;<i>op</i>&gt; {@code 0)}, where
      * &lt;<i>op</i>&gt; is one of the six comparison operators.
      *
      * @param  val BigInteger to which this BigInteger is to be compared.
      * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
      *         to, or greater than {@code val}.
      */
     public int compareTo(BigInteger val) {
         if (signum == val.signum) {
             switch (signum) {
             case 1:
                 return compareMagnitude(val);
             case -1:
                 return val.compareMagnitude(this);
             default:
                 return 0;
             }
         }
         return signum > val.signum ? 1 : -1;
     }
 
     /**
      * Compares the magnitude array of this BigInteger with the specified
      * BigInteger's. This is the version of compareTo ignoring sign.
      *
      * @param val BigInteger whose magnitude array to be compared.
      * @return -1, 0 or 1 as this magnitude array is less than, equal to or
      *         greater than the magnitude aray for the specified BigInteger's.
      */
     final int compareMagnitude(BigInteger val) {
         int[] m1 = mag;
         int len1 = m1.length;
         int[] m2 = val.mag;
         int len2 = m2.length;
         if (len1 < len2)
             return -1;
         if (len1 > len2)
             return 1;
         for (int i = 0; i < len1; i++) {
             int a = m1[i];
             int b = m2[i];
             if (a != b)
                 return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
         }
         return 0;
     }
 
     /**
      * Compares this BigInteger with the specified Object for equality.
      *
      * @param  x Object to which this BigInteger is to be compared.
      * @return {@code true} if and only if the specified Object is a
      *         BigInteger whose value is numerically equal to this BigInteger.
      */
     public boolean equals(Object x) {
         // This test is just an optimization, which may or may not help
         if (x == this)
             return true;
 
         if (!(x instanceof BigInteger))
             return false;
 
         BigInteger xInt = (BigInteger) x;
         if (xInt.signum != signum)
             return false;
 
         int[] m = mag;
         int len = m.length;
         int[] xm = xInt.mag;
         if (len != xm.length)
             return false;
 
         for (int i = 0; i < len; i++)
             if (xm[i] != m[i])
                 return false;
 
         return true;
     }
 
     /**
      * Returns the minimum of this BigInteger and {@code val}.
      *
      * @param  val value with which the minimum is to be computed.
      * @return the BigInteger whose value is the lesser of this BigInteger and
      *         {@code val}.  If they are equal, either may be returned.
      */
     public BigInteger min(BigInteger val) {
         return (compareTo(val)<0 ? this : val);
     }
 
     /**
      * Returns the maximum of this BigInteger and {@code val}.
      *
      * @param  val value with which the maximum is to be computed.
      * @return the BigInteger whose value is the greater of this and
      *         {@code val}.  If they are equal, either may be returned.
      */
     public BigInteger max(BigInteger val) {
         return (compareTo(val)>0 ? this : val);
     }
 
 
     // Hash Function
 
     /**
      * Returns the hash code for this BigInteger.
      *
      * @return hash code for this BigInteger.
      */
     public int hashCode() {
         int hashCode = 0;
 
         for (int i=0; i<mag.length; i++)
             hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
 
         return hashCode * signum;
     }
 
     /**
      * Returns the String representation of this BigInteger in the
      * given radix.  If the radix is outside the range from {@link
      * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
      * it will default to 10 (as is the case for
      * {@code Integer.toString}).  The digit-to-character mapping
      * provided by {@code Character.forDigit} is used, and a minus
      * sign is prepended if appropriate.  (This representation is
      * compatible with the {@link #BigInteger(String, int) (String,
      * int)} constructor.)
      *
      * @param  radix  radix of the String representation.
      * @return String representation of this BigInteger in the given radix.
      * @see    Integer#toString
      * @see    Character#forDigit
      * @see    #BigInteger(java.lang.String, int)
      */
     public String toString(int radix) {
         if (signum == 0)
             return "0";
         if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
             radix = 10;
 
         // Compute upper bound on number of digit groups and allocate space
         int maxNumDigitGroups = (4*mag.length + 6)/7;
         String digitGroup[] = new String[maxNumDigitGroups];
 
         // Translate number to string, a digit group at a time
         BigInteger tmp = this.abs();
         int numGroups = 0;
         while (tmp.signum != 0) {
             BigInteger d = longRadix[radix];
 
             MutableBigInteger q = new MutableBigInteger(),
                               a = new MutableBigInteger(tmp.mag),
                               b = new MutableBigInteger(d.mag);
             MutableBigInteger r = a.divide(b, q);
             BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
             BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);
 
             digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
             tmp = q2;
         }
 
         // Put sign (if any) and first digit group into result buffer
         StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
         if (signum<0)
             buf.append('-');
         buf.append(digitGroup[numGroups-1]);
 
         // Append remaining digit groups padded with leading zeros
         for (int i=numGroups-2; i>=0; i--) {
             // Prepend (any) leading zeros for this digit group
             int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
             if (numLeadingZeros != 0)
                 buf.append(zeros[numLeadingZeros]);
             buf.append(digitGroup[i]);
         }
         return buf.toString();
     }
 
     /* zero[i] is a string of i consecutive zeros. */
     private static String zeros[] = new String[64];
     static {
         zeros[63] =
             "000000000000000000000000000000000000000000000000000000000000000";
         for (int i=0; i<63; i++)
             zeros[i] = zeros[63].substring(0, i);
     }
 
     /**
      * Returns the decimal String representation of this BigInteger.
      * The digit-to-character mapping provided by
      * {@code Character.forDigit} is used, and a minus sign is
      * prepended if appropriate.  (This representation is compatible
      * with the {@link #BigInteger(String) (String)} constructor, and
      * allows for String concatenation with Java's + operator.)
      *
      * @return decimal String representation of this BigInteger.
      * @see    Character#forDigit
      * @see    #BigInteger(java.lang.String)
      */
     public String toString() {
         return toString(10);
     }
 
     /**
      * Returns a byte array containing the two's-complement
      * representation of this BigInteger.  The byte array will be in
      * <i>big-endian</i> byte-order: the most significant byte is in
      * the zeroth element.  The array will contain the minimum number
      * of bytes required to represent this BigInteger, including at
      * least one sign bit, which is {@code (ceil((this.bitLength() +
      * 1)/8))}.  (This representation is compatible with the
      * {@link #BigInteger(byte[]) (byte[])} constructor.)
      *
      * @return a byte array containing the two's-complement representation of
      *         this BigInteger.
      * @see    #BigInteger(byte[])
      */
     public byte[] toByteArray() {
         int byteLen = bitLength()/8 + 1;
         byte[] byteArray = new byte[byteLen];
 
         for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i>=0; i--) {
             if (bytesCopied == 4) {
                 nextInt = getInt(intIndex++);
                 bytesCopied = 1;
             } else {
                 nextInt >>>= 8;
                 bytesCopied++;
             }
             byteArray[i] = (byte)nextInt;
         }
         return byteArray;
     }
 
     /**
      * Converts this BigInteger to an {@code int}.  This
      * conversion is analogous to a
      * <i>narrowing primitive conversion</i> from {@code long} to
      * {@code int} as defined in section 5.1.3 of
      * <cite>The Java&trade; Language Specification</cite>:
      * if this BigInteger is too big to fit in an
      * {@code int}, only the low-order 32 bits are returned.
      * Note that this conversion can lose information about the
      * overall magnitude of the BigInteger value as well as return a
      * result with the opposite sign.
      *
      * @return this BigInteger converted to an {@code int}.
      */
     public int intValue() {
         int result = 0;
         result = getInt(0);
         return result;
     }
 
     /**
      * Converts this BigInteger to a {@code long}.  This
      * conversion is analogous to a
      * <i>narrowing primitive conversion</i> from {@code long} to
      * {@code int} as defined in section 5.1.3 of
      * <cite>The Java&trade; Language Specification</cite>:
      * if this BigInteger is too big to fit in a
      * {@code long}, only the low-order 64 bits are returned.
      * Note that this conversion can lose information about the
      * overall magnitude of the BigInteger value as well as return a
      * result with the opposite sign.
      *
      * @return this BigInteger converted to a {@code long}.
      */
     public long longValue() {
         long result = 0;
 
         for (int i=1; i>=0; i--)
             result = (result << 32) + (getInt(i) & LONG_MASK);
         return result;
     }
 
     /**
      * Converts this BigInteger to a {@code float}.  This
      * conversion is similar to the
      * <i>narrowing primitive conversion</i> from {@code double} to
      * {@code float} as defined in section 5.1.3 of
      * <cite>The Java&trade; Language Specification</cite>:
      * if this BigInteger has too great a magnitude
      * to represent as a {@code float}, it will be converted to
      * {@link Float#NEGATIVE_INFINITY} or {@link
      * Float#POSITIVE_INFINITY} as appropriate.  Note that even when
      * the return value is finite, this conversion can lose
      * information about the precision of the BigInteger value.
      *
      * @return this BigInteger converted to a {@code float}.
      */
     public float floatValue() {
         // Somewhat inefficient, but guaranteed to work.
         return Float.parseFloat(this.toString());
     }
 
     /**
      * Converts this BigInteger to a {@code double}.  This
      * conversion is similar to the
      * <i>narrowing primitive conversion</i> from {@code double} to
      * {@code float} as defined in section 5.1.3 of
      * <cite>The Java&trade; Language Specification</cite>:
      * if this BigInteger has too great a magnitude
      * to represent as a {@code double}, it will be converted to
      * {@link Double#NEGATIVE_INFINITY} or {@link
      * Double#POSITIVE_INFINITY} as appropriate.  Note that even when
      * the return value is finite, this conversion can lose
      * information about the precision of the BigInteger value.
      *
      * @return this BigInteger converted to a {@code double}.
      */
     public double doubleValue() {
         // Somewhat inefficient, but guaranteed to work.
         return Double.parseDouble(this.toString());
     }
 
     /**
      * Returns a copy of the input array stripped of any leading zero bytes.
      */
     private static int[] stripLeadingZeroInts(int val[]) {
         int vlen = val.length;
         int keep;
 
         // Find first nonzero byte
         for (keep = 0; keep < vlen && val[keep] == 0; keep++)
             ;
         return java.util.Arrays.copyOfRange(val, keep, vlen);
     }
 
     /**
      * Returns the input array stripped of any leading zero bytes.
      * Since the source is trusted the copying may be skipped.
      */
     private static int[] trustedStripLeadingZeroInts(int val[]) {
         int vlen = val.length;
         int keep;
 
         // Find first nonzero byte
         for (keep = 0; keep < vlen && val[keep] == 0; keep++)
             ;
         return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
     }
 
     /**
      * Returns a copy of the input array stripped of any leading zero bytes.
      */
     private static int[] stripLeadingZeroBytes(byte a[]) {
         int byteLength = a.length;
         int keep;
 
         // Find first nonzero byte
         for (keep = 0; keep < byteLength && a[keep]==0; keep++)
             ;
 
         // Allocate new array and copy relevant part of input array
         int intLength = ((byteLength - keep) + 3) >>> 2;
         int[] result = new int[intLength];
         int b = byteLength - 1;
         for (int i = intLength-1; i >= 0; i--) {
             result[i] = a[b--] & 0xff;
             int bytesRemaining = b - keep + 1;
             int bytesToTransfer = Math.min(3, bytesRemaining);
             for (int j=8; j <= (bytesToTransfer << 3); j += 8)
                 result[i] |= ((a[b--] & 0xff) << j);
         }
         return result;
     }
 
     /**
      * Takes an array a representing a negative 2's-complement number and
      * returns the minimal (no leading zero bytes) unsigned whose value is -a.
      */
     private static int[] makePositive(byte a[]) {
         int keep, k;
         int byteLength = a.length;
 
         // Find first non-sign (0xff) byte of input
         for (keep=0; keep<byteLength && a[keep]==-1; keep++)
             ;
 
 
         /* Allocate output array.  If all non-sign bytes are 0x00, we must
          * allocate space for one extra output byte. */
         for (k=keep; k<byteLength && a[k]==0; k++)
             ;
 
         int extraByte = (k==byteLength) ? 1 : 0;
         int intLength = ((byteLength - keep + extraByte) + 3)/4;
         int result[] = new int[intLength];
 
         /* Copy one's complement of input into output, leaving extra
          * byte (if it exists) == 0x00 */
         int b = byteLength - 1;
         for (int i = intLength-1; i >= 0; i--) {
             result[i] = a[b--] & 0xff;
             int numBytesToTransfer = Math.min(3, b-keep+1);
             if (numBytesToTransfer < 0)
                 numBytesToTransfer = 0;
             for (int j=8; j <= 8*numBytesToTransfer; j += 8)
                 result[i] |= ((a[b--] & 0xff) << j);
 
             // Mask indicates which bits must be complemented
             int mask = -1 >>> (8*(3-numBytesToTransfer));
             result[i] = ~result[i] & mask;
         }
 
         // Add one to one's complement to generate two's complement
         for (int i=result.length-1; i>=0; i--) {
             result[i] = (int)((result[i] & LONG_MASK) + 1);
             if (result[i] != 0)
                 break;
         }
 
         return result;
     }
 
     /**
      * Takes an array a representing a negative 2's-complement number and
      * returns the minimal (no leading zero ints) unsigned whose value is -a.
      */
     private static int[] makePositive(int a[]) {
         int keep, j;
 
         // Find first non-sign (0xffffffff) int of input
         for (keep=0; keep<a.length && a[keep]==-1; keep++)
             ;
 
         /* Allocate output array.  If all non-sign ints are 0x00, we must
          * allocate space for one extra output int. */
         for (j=keep; j<a.length && a[j]==0; j++)
             ;
         int extraInt = (j==a.length ? 1 : 0);
         int result[] = new int[a.length - keep + extraInt];
 
         /* Copy one's complement of input into output, leaving extra
          * int (if it exists) == 0x00 */
         for (int i = keep; i<a.length; i++)
             result[i - keep + extraInt] = ~a[i];
 
         // Add one to one's complement to generate two's complement
         for (int i=result.length-1; ++result[i]==0; i--)
             ;
 
         return result;
     }
 
     /*
      * The following two arrays are used for fast String conversions.  Both
      * are indexed by radix.  The first is the number of digits of the given
      * radix that can fit in a Java long without "going negative", i.e., the
      * highest integer n such that radix**n < 2**63.  The second is the
      * "long radix" that tears each number into "long digits", each of which
      * consists of the number of digits in the corresponding element in
      * digitsPerLong (longRadix[i] = i**digitPerLong[i]).  Both arrays have
      * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
      * used.
      */
     private static int digitsPerLong[] = {0, 0,
         62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
         14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};
 
     private static BigInteger longRadix[] = {null, null,
         valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
         valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
         valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
         valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
         valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
         valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
         valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
         valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
         valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
         valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
         valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
         valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
         valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
         valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
         valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
         valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
         valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
         valueOf(0x41c21cb8e1000000L)};
 
     /*
      * These two arrays are the integer analogue of above.
      */
     private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
         11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
         6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};
 
     private static int intRadix[] = {0, 0,
         0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
         0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
         0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f,  0x10000000,
         0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
         0x6c20a40,  0x8d2d931,  0xb640000,  0xe8d4a51,  0x1269ae40,
         0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
         0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
     };
 
     /**
      * These routines provide access to the two's complement representation
      * of BigIntegers.
      */
 
     /**
      * Returns the length of the two's complement representation in ints,
      * including space for at least one sign bit.
      */
     private int intLength() {
         return (bitLength() >>> 5) + 1;
     }
 
     /* Returns sign bit */
     private int signBit() {
         return signum < 0 ? 1 : 0;
     }
 
     /* Returns an int of sign bits */
     private int signInt() {
         return signum < 0 ? -1 : 0;
     }
 
     /**
      * Returns the specified int of the little-endian two's complement
      * representation (int 0 is the least significant).  The int number can
      * be arbitrarily high (values are logically preceded by infinitely many
      * sign ints).
      */
     private int getInt(int n) {
         if (n < 0)
             return 0;
         if (n >= mag.length)
             return signInt();
 
         int magInt = mag[mag.length-n-1];
 
         return (signum >= 0 ? magInt :
                 (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
     }
 
     /**
      * Returns the index of the int that contains the first nonzero int in the
      * little-endian binary representation of the magnitude (int 0 is the
      * least significant). If the magnitude is zero, return value is undefined.
      */
      private int firstNonzeroIntNum() {
          int fn = firstNonzeroIntNum - 2;
          if (fn == -2) { // firstNonzeroIntNum not initialized yet
              fn = 0;
 
              // Search for the first nonzero int
              int i;
              int mlen = mag.length;
              for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
                  ;
              fn = mlen - i - 1;
              firstNonzeroIntNum = fn + 2; // offset by two to initialize
          }
          return fn;
      }
 
     /** use serialVersionUID from JDK 1.1. for interoperability */
     private static final long serialVersionUID = -8287574255936472291L;
 
     /**
      * Serializable fields for BigInteger.
      *
      * @serialField signum  int
      *              signum of this BigInteger.
      * @serialField magnitude int[]
      *              magnitude array of this BigInteger.
      * @serialField bitCount  int
      *              number of bits in this BigInteger
      * @serialField bitLength int
      *              the number of bits in the minimal two's-complement
      *              representation of this BigInteger
      * @serialField lowestSetBit int
      *              lowest set bit in the twos complement representation
      */
     private static final ObjectStreamField[] serialPersistentFields = {
         new ObjectStreamField("signum", Integer.TYPE),
         new ObjectStreamField("magnitude", byte[].class),
         new ObjectStreamField("bitCount", Integer.TYPE),
         new ObjectStreamField("bitLength", Integer.TYPE),
         new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
         new ObjectStreamField("lowestSetBit", Integer.TYPE)
         };
 
     /**
      * Reconstitute the {@code BigInteger} instance from a stream (that is,
      * deserialize it). The magnitude is read in as an array of bytes
      * for historical reasons, but it is converted to an array of ints
      * and the byte array is discarded.
      * Note:
      * The current convention is to initialize the cache fields, bitCount,
      * bitLength and lowestSetBit, to 0 rather than some other marker value.
      * Therefore, no explicit action to set these fields needs to be taken in
      * readObject because those fields already have a 0 value be default since
      * defaultReadObject is not being used.
      */
     private void readObject(java.io.ObjectInputStream s)
         throws java.io.IOException, ClassNotFoundException {
         /*
          * In order to maintain compatibility with previous serialized forms,
          * the magnitude of a BigInteger is serialized as an array of bytes.
          * The magnitude field is used as a temporary store for the byte array
          * that is deserialized. The cached computation fields should be
          * transient but are serialized for compatibility reasons.
          */
 
         // prepare to read the alternate persistent fields
         ObjectInputStream.GetField fields = s.readFields();
 
         // Read the alternate persistent fields that we care about
         int sign = fields.get("signum", -2);
         byte[] magnitude = (byte[])fields.get("magnitude", null);
 
         // Validate signum
         if (sign < -1 || sign > 1) {
             String message = "BigInteger: Invalid signum value";
             if (fields.defaulted("signum"))
                 message = "BigInteger: Signum not present in stream";
             throw new java.io.StreamCorruptedException(message);
         }
         if ((magnitude.length == 0) != (sign == 0)) {
             String message = "BigInteger: signum-magnitude mismatch";
             if (fields.defaulted("magnitude"))
                 message = "BigInteger: Magnitude not present in stream";
             throw new java.io.StreamCorruptedException(message);
         }
 
         // Commit final fields via Unsafe
         unsafe.putIntVolatile(this, signumOffset, sign);
 
         // Calculate mag field from magnitude and discard magnitude
         unsafe.putObjectVolatile(this, magOffset,
                                  stripLeadingZeroBytes(magnitude));
     }
 
     // Support for resetting final fields while deserializing
     private static final sun.misc.Unsafe unsafe = sun.misc.Unsafe.getUnsafe();
     private static final long signumOffset;
     private static final long magOffset;
     static {
         try {
             signumOffset = unsafe.objectFieldOffset
                 (BigInteger.class.getDeclaredField("signum"));
             magOffset = unsafe.objectFieldOffset
                 (BigInteger.class.getDeclaredField("mag"));
         } catch (Exception ex) {
             throw new Error(ex);
         }
     }
 
     /**
      * Save the {@code BigInteger} instance to a stream.
      * The magnitude of a BigInteger is serialized as a byte array for
      * historical reasons.
      *
      * @serialData two necessary fields are written as well as obsolete
      *             fields for compatibility with older versions.
      */
     private void writeObject(ObjectOutputStream s) throws IOException {
         // set the values of the Serializable fields
         ObjectOutputStream.PutField fields = s.putFields();
         fields.put("signum", signum);
         fields.put("magnitude", magSerializedForm());
         // The values written for cached fields are compatible with older
         // versions, but are ignored in readObject so don't otherwise matter.
         fields.put("bitCount", -1);
         fields.put("bitLength", -1);
         fields.put("lowestSetBit", -2);
         fields.put("firstNonzeroByteNum", -2);
 
         // save them
         s.writeFields();
 }
 
     /**
      * Returns the mag array as an array of bytes.
      */
     private byte[] magSerializedForm() {
         int len = mag.length;
 
         int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
         int byteLen = (bitLen + 7) >>> 3;
         byte[] result = new byte[byteLen];
 
         for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
              i>=0; i--) {
             if (bytesCopied == 4) {
                 nextInt = mag[intIndex--];
                 bytesCopied = 1;
             } else {
                 nextInt >>>= 8;
                 bytesCopied++;
             }
             result[i] = (byte)nextInt;
         }
         return result;
     }
 }