/*

  * Copyright (c) 1996, 2007, Oracle and/or its affiliates. All rights reserved.

  * 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

  * or visit www.oracle.com if you need additional information or have any

  * questions.

  */

 /*

  * 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} ≤ 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} ≤ 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} ≤ 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 ≤ 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]);