#include <vector>
#include <algorithm>
#include <cstring>
#include <iterator>
#include <stdexcept>
#include <cstdint>
#include <cassert>
using std::uint64_t;
class BigInteger{
    const int signum;
    std::vector<int> mag;   
    int bitCount;   
    int bitLength;
    int lowestSetBit;
    int firstNonzeroIntNum;
    const static uint64_t LONG_MASK = 0xffffffffL;
    static const int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26)
    static const int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000;
    static const int KARATSUBA_THRESHOLD = 80;
    static const int TOOM_COOK_THRESHOLD = 240;
    static const int KARATSUBA_SQUARE_THRESHOLD = 128;
    static const int TOOM_COOK_SQUARE_THRESHOLD = 216;
    static const int BURNIKEL_ZIEGLER_THRESHOLD = 80;
    static const int BURNIKEL_ZIEGLER_OFFSET = 40;
    static const int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20;
    static const int MULTIPLY_SQUARE_THRESHOLD = 20;
    static const int MONTGOMERY_INTRINSIC_THRESHOLD = 512;
    BigInteger(std::vector<char> val) {
        assert(val.size() != 0);
        if (val[0] < 0) {
            mag = makePositive(val);
            signum = -1;
        } else {
            mag = stripLeadingZeroBytes(val);
            signum = (mag.length == 0 ? 0 : 1);
        }
        if (mag.size() >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    
    BigInteger(std::vector<int> val) {
        assert(val.size() != 0);

        if (val[0] < 0) {
            mag = makePositive(val);
            signum = -1;
        } else {
            mag = trustedStripLeadingZeroInts(val);
            signum = (mag.size() == 0 ? 0 : 1);
        }
        if (mag.size() >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }
     
    
    BigInteger(int signum, std::vector<char> magnitude) {
        mag = stripLeadingZeroBytes(magnitude);

        assert(!(signum < -1 || signum > 1));

        if (mag.size() == 0) {
            signum = 0;
        } else {
            assert(signum != 0);
            signum = signum;
        }
        if (mag.size() >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    
    BigInteger(int signum, std::vector<int> magnitude) {
        mag = stripLeadingZeroInts(magnitude);

        assert(!(signum < -1 || signum > 1));

        if (this.mag.length == 0) {
            signum = 0;
        } else {
            assert(signum != 0);
            signum = signum;
        }
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    

    /*
     * Constructs a new BigInteger using a char array with radix=10.
     * Sign is precalculated outside and not allowed in the val.
     */
    BigInteger(std::vector<char> val, int sign, int len) {
        int cursor = 0, numDigits;

        // 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
        unsigned int numWords;
        if (len < 10) {
            numWords = 1;
        } else {
            uint64_t numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1;
            if (numBits + 31 >= (1L << 32)) {
                reportOverflow();
            }
            numWords = (int) (numBits + 31) >>> 5;
        }
        std::vector<int> magnitude(numBits);

        // 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);
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }
	int digit(char a){
		assert((int)(a - '0') < 10 && (int)(a - '0') >= 0);
		return (int)(a - '0');
	}
    // 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.
    int parseInt(const std::vector<char>& source, int start, int end) {
        int result = digit(source[start++]);
		
        for (int index = start; index < end; index++) {
            int nextVal = digit(source[index]);
            result = 10 * result + nextVal;
        }

        return result;
    }

    // bitsPerDigit in the given radix times 1024
    // Rounded up to avoid underallocation.
    static uint64_t 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
    static void destructiveMulAdd(std::vector<int>& x, int y, int z) {
        // Perform the multiplication word by word
        uint64_t ylong = y & LONG_MASK;
        uint64_t zlong = z & LONG_MASK;
        int len = x.length;

        uint64_t product = 0;
        uint64_t 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
        uint64_t 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;
        }
    }

    
    BigInteger(std::string val) : BigInteger(val, 10) {
        
    }

    
    BigInteger(int numBits, std::mt19937_64& rnd) : BigInteger(1, randomBits(numBits, rnd)) {
        
    }

    static std::vector<char> randomBits(unsigned int numBits, std::mt19937_64& rnd) {
        unsigned int numBytes = (unsigned int)(((uint64_t)numBits+7)/8); // avoid overflow
        std::vector<char> randomBits(numBytes);
		std::uniform_int_distribution<char> dis(-128,127);
        // Generate random bytes and mask out any excess bits
        if (numBytes > 0) {
            std::generate(randomBits.begin(), randomBits.end(), [&](){return dis(rnd)});
            int excessBits = 8*numBytes - numBits;
            randomBits[0] &= (1 << (8-excessBits)) - 1;
        }
        return randomBits;
    }

    
    BigInteger(int bitLength, int certainty, std::mt19937_64& rnd) {
        BigInteger prime;

        assert(bitLength >= 2);
        prime = (bitLength < SMALL_PRIME_THRESHOLD
                                ? 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.
    // The cutoff of 95 was chosen empirically for best performance.
    static const int SMALL_PRIME_THRESHOLD = 95;

    // Certainty required to meet the spec of probablePrime
    static const int DEFAULT_PRIME_CERTAINTY = 100;

    
    /*static BigInteger probablePrime(int bitLength, Random rnd) {
        if (bitLength < 2)
            throw new ArithmeticException("bitLength < 2");

        return (bitLength < SMALL_PRIME_THRESHOLD ?
                smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
                largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
    }

    
    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;
        }
    }

    static const BigInteger SMALL_PRIME_PRODUCT
                       = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);

    
    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 = getPrimeSearchLen(bitLength);
        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;
    }

   
    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 = getPrimeSearchLen(result.bitLength());

        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));
        }
    }

    static int getPrimeSearchLen(int bitLength) {
        if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) {
            throw new ArithmeticException("Prime search implementation restriction on bitLength");
        }
        return bitLength / 20 * 64;
    }*/

    
    bool primeToCertainty(int certainty, std::mt19937_64& random) {
        int rounds = 0;
        int n = (std::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 = 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();
    }

    
    boolean passesLucasLehmer() {
        BigInteger thisPlusOne = this->add(ONE);

        // Step 1
        int d = 5;
        while (jacobiSymbol(d, this) != -1) {
            // 5, -7, 9, -11, ...
            d = (d < 0) ? std::abs(d)+2 : -(d+2);
        }

        // Step 2
        BigInteger u = lucasLehmerSequence(d, thisPlusOne, *this);

        // Step 3
        return u.mod(*this).equals(ZERO);
    }

    
    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;
    }

    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;
    }

    
    bool passesMillerRabin(int iterations, std::mt19937_64& 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
        for (int i=0; i < iterations; i++) {
            // Generate a uniform random on (1, this)
            BigInteger b;
            do {
                b = 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;
    }

    
    BigInteger(const std::vector<int> magnitude&, int signum) {
        this.signum = (magnitude.siz() == 0 ? 0 : signum);
        this.mag = magnitude;
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    
    BigInteger(const std::vector<int> magnitude&e, int signum) {
        signum = (magnitude.length == 0 ? 0 : signum);
        mag = stripLeadingZeroBytes(magnitude);
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    
    void checkRange() {
        if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) {
            reportOverflow();
        }
    }

    static void reportOverflow() {
        std::cout << "BigInteger would overflow supported range" << std::endl;
		throw 1;
    }

    //Static Factory Methods

    
    static BigInteger valueOf(std::int64_t 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 BigInteger(val);
    }

    
    BigInteger(std::int64_t val) {
        if (val < 0) {
            val = -val;
            signum = -1;
        } else {
            signum = 1;
        }

        int highWord = (int)(((uint64_t)val) >> 32);
        if (highWord == 0) {
            mag = std::vector<int>(1);
            mag[0] = (int)val;
        } else {
            mag = std::vector<int>(2);
            mag[0] = highWord;
            mag[1] = (int)val;
        }
    }

    
    static BigInteger valueOf(std::vector<int> val) {
        return (val[0] > 0 ? BigInteger(val, 1) : BigInteger(val));
    }

    // Constants

    
    const static int MAX_CONSTANT = 16;
    static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
    static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];

    
    static volatile BigInteger[][] powerCache;

    
    static const double[] logCache;

    
    static const double LOG_TWO = Math.log(2.0);

    static void initStuff(){
        for (int i = 1; i <= MAX_CONSTANT; i++) {
            std::vector<int> magnitude(1);
            magnitude[0] = i;
            posConst[i] = BigInteger(magnitude,  1);
            negConst[i] = BigInteger(magnitude, -1);
        }

        /*
         * Initialize the cache of radix^(2^x) values used for base conversion
         * with just the very first value.  Additional values will be created
         * on demand.
         */
        powerCache = new BigInteger[Character.MAX_RADIX+1][];
        logCache = new double[Character.MAX_RADIX+1];

        for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
            powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) };
            logCache[i] = Math.log(i);
        }
    }

    
    static const BigInteger ZERO = new BigInteger(std::vector<int>(0), 0);

    
    static const BigInteger ONE = valueOf(1);

    
    static const BigInteger TWO = valueOf(2);

    
    static const BigInteger NEGATIVE_ONE = valueOf(-1);

    
    static const BigInteger TEN = valueOf(10);

    // Arithmetic Operations

    
    BigInteger add(BigInteger val) {
        if (val.signum == 0)
            return *this;
        if (signum == 0)
            return val;
        if (val.signum == signum)
            return BigInteger(add(mag, val.mag), signum);

        int cmp = compareMagnitude(val);
        if (cmp == 0)
            return ZERO;
        std::vector<int> resultMag = (cmp > 0 ? subtract(mag, val.mag)
                           : subtract(val.mag, mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);

        return BigInteger(resultMag, cmp == signum ? 1 : -1);
    }
	static int signum(std::int64_t a){
		if(a == 0)return 0;
		if(a > 0)return 1;
		return -1;
	}
    
    BigInteger add(std::int64_t val) {
        if (val == 0)
            return *this;
        if (signum == 0)
            return valueOf(val);
        if (signum(val) == signum)
            return BigInteger(add(mag, Math.abs(val)), signum);
        int cmp = compareMagnitude(val);
        if (cmp == 0)
            return ZERO;
        std::vector<int> resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);
        return BigInteger(resultMag, cmp == signum ? 1 : -1);
    }

    
    static std::vector<int> add(std::vector<int> x, std::uint64_t val) {
        std::vector<int> y;
        long sum = 0;
        int xIndex = x.size();
        std::vector<int> result;
        int highWord = (int)(val >>> 32);
        if (highWord == 0) {
            result = std::vector<int>(xIndex);
            sum = (x[--xIndex] & LONG_MASK) + val;
            result[xIndex] = (int)sum;
        } else {
            if (xIndex == 1) {
                result = std::vector<int>(2);
                sum = val  + (x[0] & LONG_MASK);
                result[1] = (int)sum;
                result[0] = (int)(sum >>> 32);
                return result;
            } else {
                result = std::vector<int>(xIndex);
                sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
                result[xIndex] = (int)sum;
                sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
                result[xIndex] = (int)sum;
            }
        }
        // Copy remainder of longer number while carry propagation is required
        bool 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) {
            std::vector<int> bigger(result.length + 1);
            //System.arraycopy(result, 0, bigger, 1, result.length);
			std::copy(result.begin(),result.end(), bigger.begin());
            bigger[0] = 0x01;
            return bigger;
        }
        return result;
    }

    
    static std::vector<int> add(std::vector<int> x, std::vector<int> y) {
        // If x is shorter, swap the two arrays
        if (x.size() < y.size()) {
            std::swap(x,y);
			/*int[] tmp = x;
            x = y;
            y = tmp;*/
        }

        int xIndex = x.length;
        int yIndex = y.length;
        std::vector<int> result(xIndex);// = new int[xIndex];
        long sum = 0;
        if (yIndex == 1) {
            sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
            result[xIndex] = (int)sum;
        } else {
            // 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) {
            std::vector<int> bigger(result.length + 1);
            std::copy(result.begin(),result.end(),bigger.begin());
            bigger[0] = 0x01;
            return bigger;
        }
        return result;
    }

    static int[] subtract(std::int64_t val, int[] little) {
        int highWord = (int)(((uint64_t)val) >> 32);
        if (highWord == 0) {
            std::vector<int>result(1);
            result[0] = (int)(val - (little[0] & LONG_MASK));
            return result;
        } else {
            std::vector<int> result(2);
            if (little.length == 1) {
                std::int64_t difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
                result[1] = (int)difference;
                // Subtract remainder of longer number while borrow propagates
                bool borrow = (difference >> 32 != 0);
                if (borrow) {
                    result[0] = highWord - 1;
                } else {        // Copy remainder of longer number
                    result[0] = highWord;
                }
                return result;
            } else { // little.length == 2
                std::int64_t difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
                result[1] = (int)difference;
                difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
                result[0] = (int)difference;
                return result;
            }
        }
    }

    
    static std::vector<int> subtract(std::vector<int> big, std::int64_t val) {
        int highWord = (int)(((uint64_t)val) >> 32);
        int bigIndex = big.size();
        std::vector<int> result(bigIndex);
        long difference = 0;

        if (highWord == 0) {
            difference = (big[--bigIndex] & LONG_MASK) - val;
            result[bigIndex] = (int)difference;
        } else {
            difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
            result[bigIndex] = (int)difference;
            difference = (big[--bigIndex] & LONG_MASK) - (highWord & 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;
    }

    
    BigInteger subtract(BigInteger val) {
        if (val.signum == 0)
            return *this;
        if (signum == 0)
            return val.negate();
        if (val.signum != signum)
            return BigInteger(add(mag, val.mag), signum);

        int cmp = compareMagnitude(val);
        if (cmp == 0)
            return ZERO;
        std::vector<int> resultMag = (cmp > 0 ? subtract(mag, val.mag)
                           : subtract(val.mag, mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);
        return BigInteger(resultMag, cmp == signum ? 1 : -1);
    }

    
    static std::vector<int> subtract(std::vector<int> big, std::vector<int> little) {
        int bigIndex = big.size();
        std::vector<int> result(bigIndex);
        int littleIndex = little.size();
        std::int64_t 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
        bool 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;
    }

    
    BigInteger multiply(BigInteger val) {
        if (val.signum == 0 || signum == 0)
            return ZERO;

        int xlen = mag.size();

        if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) {
            return square();
        }

        int ylen = val.mag.size();

        if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
            int resultSign = signum == val.signum ? 1 : -1;
            if (val.mag.size() == 1) {
                return multiplyByInt(mag,val.mag[0], resultSign);
            }
            if (mag.size() == 1) {
                return multiplyByInt(val.mag,mag[0], resultSign);
            }
            std::vector<int> result = multiplyToLen(mag, xlen,
                                         val.mag, ylen, null);
            result = trustedStripLeadingZeroInts(result);
            return BigInteger(result, resultSign);
        } else {
            if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
                return multiplyKaratsuba(this, val);
            } else {
                return multiplyToomCook3(this, val);
            }
        }
    }

    static BigInteger multiplyByInt(std::vector<int> x, int y, int sign) {
        if (__builtin_popcount(y) == 1) {
            return BigInteger(shiftLeft(x,__builtin_ctz(y)), sign);
        }
        int xlen = x.length;
        std::vector<int> rmag(xlen + 1);
        long carry = 0;
        long yl = y & LONG_MASK;
        int rstart = rmag.length - 1;
        for (int i = xlen - 1; i >= 0; i--) {
            std::uint64_t product = (x[i] & LONG_MASK) * yl + carry;
            rmag[rstart--] = (int)product;
            carry = product >> 32;
        }
        if (carry == 0L) {
            rmag.erase(rmag.begin());
        } else {
            rmag[rstart] = (int)carry;
        }
        return BigInteger(rmag, sign);
    }

    
    BigInteger multiply(std::uint64_t v) {
        if (v == 0 || signum == 0)
          return ZERO;
        if (v == std::numeric_limits<std::uint64_t>::max())
            return multiply(BigInteger.valueOf(v));
        int rsign = (v > 0 ? signum : -signum);
        if (v < 0)
            v = -v;
        uint64_t dh = v >> 32;      // higher order bits
        uint64_t dl = v & LONG_MASK; // lower order bits

        int xlen = mag.length;
        std::vector<int> value = mag;
        std::vector<int> rmag = (dh == 0L) ? (std::vector<int>([xlen + 1])) : (std::vector<int>([xlen + 2]));
        uint64_t carry = 0;
        int rstart = rmag.length - 1;
        for (int i = xlen - 1; i >= 0; i--) {
            uint64_t 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 == 0LL)
            rmag.erase(rmag.begin());
        return BigInteger(rmag, rsign);
    }

    
    static std::vector<int> multiplyToLen(std::vector<int> x, int xlen, std::vector<int> y, int ylen, std::vector<int> z) {
        int xstart = xlen - 1;
        int ystart = ylen - 1;

        if (z.length < (xlen+ ylen))
            z = std::vector<int>(xlen+ylen);

        uint64_t 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;
    }

    
    static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
        int xlen = x.mag.size();
        int ylen = y.mag.size();

        // The number of ints in each half of the number.
        int half = (std::max(xlen, ylen)+1) / 2;

        // xl and yl are the lower halves of x and y respectively,
        // xh and yh are the upper halves.
        BigInteger xl = x.getLower(half);
        BigInteger xh = x.getUpper(half);
        BigInteger yl = y.getLower(half);
        BigInteger yh = y.getUpper(half);

        BigInteger p1 = xh.multiply(yh);  // p1 = xh*yh
        BigInteger p2 = xl.multiply(yl);  // p2 = xl*yl

        // p3=(xh+xl)*(yh+yl)
        BigInteger p3 = xh.add(xl).multiply(yh.add(yl));

        // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
        BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);

        if (x.signum != y.signum) {
            return result.negate();
        } else {
            return result;
        }
    }

    
    static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
        int alen = a.mag.size();
        int blen = b.mag.size();

        int largest = std::max(alen, blen);

        // k is the size (in ints) of the lower-order slices.
        int k = (largest+2)/3;   // Equal to ceil(largest/3)

        // r is the size (in ints) of the highest-order slice.
        int r = largest - 2*k;

        // Obtain slices of the numbers. a2 and b2 are the most significant
        // bits of the numbers a and b, and a0 and b0 the least significant.
        BigInteger a0, a1, a2, b0, b1, b2;
        a2 = a.getToomSlice(k, r, 0, largest);
        a1 = a.getToomSlice(k, r, 1, largest);
        a0 = a.getToomSlice(k, r, 2, largest);
        b2 = b.getToomSlice(k, r, 0, largest);
        b1 = b.getToomSlice(k, r, 1, largest);
        b0 = b.getToomSlice(k, r, 2, largest);

        BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;

        v0 = a0.multiply(b0);
        da1 = a2.add(a0);
        db1 = b2.add(b0);
        vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
        da1 = da1.add(a1);
        db1 = db1.add(b1);
        v1 = da1.multiply(db1);
        v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
             db1.add(b2).shiftLeft(1).subtract(b0));
        vinf = a2.multiply(b2);

        // The algorithm requires two divisions by 2 and one by 3.
        // All divisions are known to be exact, that is, they do not produce
        // remainders, and all results are positive.  The divisions by 2 are
        // implemented as right shifts which are relatively efficient, leaving
        // only an exact division by 3, which is done by a specialized
        // linear-time algorithm.
        t2 = v2.subtract(vm1).exactDivideBy3();
        tm1 = v1.subtract(vm1).shiftRight(1);
        t1 = v1.subtract(v0);
        t2 = t2.subtract(t1).shiftRight(1);
        t1 = t1.subtract(tm1).subtract(vinf);
        t2 = t2.subtract(vinf.shiftLeft(1));
        tm1 = tm1.subtract(t2);

        // Number of bits to shift left.
        int ss = k*32;

        BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);

        if (a.signum != b.signum) {
            return result.negate();
        } else {
            return result;
        }
    }


    
    BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
                                    int fullsize) {
        int start, end, sliceSize, len, offset;

        len = mag.size();
        offset = fullsize - len;

        if (slice == 0) {
            start = 0 - offset;
            end = upperSize - 1 - offset;
        } else {
            start = upperSize + (slice-1)*lowerSize - offset;
            end = start + lowerSize - 1;
        }

        if (start < 0) {
            start = 0;
        }
        if (end < 0) {
           return ZERO;
        }

        sliceSize = (end-start) + 1;

        if (sliceSize <= 0) {
            return ZERO;
        }

        // While performing Toom-Cook, all slices are positive and
        // the sign is adjusted when the const number is composed.
        if (start == 0 && sliceSize >= len) {
            return this->abs();
        }

        std::vector<int> intSlice(sliceSize);
        std::copy(mag.begin() + start,mag.begin() + start + sliceSize, intSlice.begin());

        return BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
    }

    
    BigInteger exactDivideBy3() {
        int len = mag.size();
        std::vector<int> result(len);
        std::int64_t x, w, q, borrow;
        borrow = 0L;
        for (int i = len-1; i >= 0; i--) {
            x = (mag[i] & LONG_MASK);
            w = x - borrow;
            if (borrow > x) {      // Did we make the number go negative?
                borrow = 1LL;
            } else {
                borrow = 0LL;
            }

            // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32).  Thus,
            // the effect of this is to divide by 3 (mod 2^32).
            // This is much faster than division on most architectures.
            q = (w * 0xAAAAAAABL) & LONG_MASK;
            result[i] = (int) q;

            // Now check the borrow. The second check can of course be
            // eliminated if the first fails.
            if (q >= 0x55555556L) {
                borrow++;
                if (q >= 0xAAAAAAABL)
                    borrow++;
            }
        }
        result = trustedStripLeadingZeroInts(result);
        return BigInteger(result, signum);
    }

    
    BigInteger getLower(int n) {
        int len = mag.size();

        if (len <= n) {
            return abs();
        }

        std::vector<int>lowerInts(n);
        std::copy(mag.begin() + (len - n), mag.end(), lowerInts.begin());

        return BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
    }

    
    BigInteger getUpper(int n) {
        int len = mag.length;

        if (len <= n) {
            return ZERO;
        }

        int upperLen = len - n;
        std::vector<int> upperInts(upperLen);
        std::copy(mag.begin(), mag.begin() + upperLen, upperInts);

        return BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
    }

    // Squaring

    
    BigInteger square() {
        if (signum == 0) {
            return ZERO;
        }
        int len = mag.size();

        if (len < KARATSUBA_SQUARE_THRESHOLD) {
            std::vector<int> z = squareToLen(mag, len, null);
            return BigInteger(trustedStripLeadingZeroInts(z), 1);
        } else {
            if (len < TOOM_COOK_SQUARE_THRESHOLD) {
                return squareKaratsuba();
            } else {
                return squareToomCook3();
            }
        }
    }

    
    static std::vector<int> squareToLen(std::vector<int> x, int len, std::vector<int> z) {
         int zlen = len << 1;
         if (z == null || z.length < zlen)
             z = std::vector<int>(zlen);

         // Execute checks before calling intrinsified method.
         implSquareToLenChecks(x, len, z, zlen);
         return implSquareToLen(x, len, z, zlen);
     }

     
     static void implSquareToLenChecks(std::vector<int> x, int len, std::vector<int> z, int zlen){
         if (len < 1) {
             throw std::invalid_argument("invalid input length: " + len);
         }
         if (len > x.size()) {
             throw std::invalid_argument("input length out of bound: " +
                                        len + " > " + x.size());
         }
         if (len * 2 > z.size()) {
             throw std::invalid_argument("input length out of bound: " +
                                        (len * 2) + " > " + z.size());
         }
         if (zlen < 1) {
             throw std::invalid_argument("invalid input length: " + zlen);
         }
         if (zlen > z.size()) {
             throw std::invalid_argument("input length out of bound: " +
                                        len + " > " + z.size());
         }
     }

     
     static std::vector<int> implSquareToLen(std::vector<int> x, int len, std::vector<int> z, int zlen) {
        /*
         * 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.
         */

        // Store the squares, right shifted one bit (i.e., divided by 2)
        int lastProductLowWord = 0;
        for (int j=0, i=0; j < len; j++) {
            std::int64_t piece = (x[j] & LONG_MASK);
            uint64_t 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;
    }

    
    BigInteger squareKaratsuba() {
        int half = (mag.size()+1) / 2;

        BigInteger xl = getLower(half);
        BigInteger xh = getUpper(half);

        BigInteger xhs = xh.square();  // xhs = xh^2
        BigInteger xls = xl.square();  // xls = xl^2

        // xh^2 << 64  +  (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
        return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
    }

    
    BigInteger squareToomCook3() {
        int len = mag.size();

        // k is the size (in ints) of the lower-order slices.
        int k = (len+2)/3;   // Equal to ceil(largest/3)

        // r is the size (in ints) of the highest-order slice.
        int r = len - 2*k;

        // Obtain slices of the numbers. a2 is the most significant
        // bits of the number, and a0 the least significant.
        BigInteger a0, a1, a2;
        a2 = getToomSlice(k, r, 0, len);
        a1 = getToomSlice(k, r, 1, len);
        a0 = getToomSlice(k, r, 2, len);
        BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;

        v0 = a0.square();
        da1 = a2.add(a0);
        vm1 = da1.subtract(a1).square();
        da1 = da1.add(a1);
        v1 = da1.square();
        vinf = a2.square();
        v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();

        // The algorithm requires two divisions by 2 and one by 3.
        // All divisions are known to be exact, that is, they do not produce
        // remainders, and all results are positive.  The divisions by 2 are
        // implemented as right shifts which are relatively efficient, leaving
        // only a division by 3.
        // The division by 3 is done by an optimized algorithm for this case.
        t2 = v2.subtract(vm1).exactDivideBy3();
        tm1 = v1.subtract(vm1).shiftRight(1);
        t1 = v1.subtract(v0);
        t2 = t2.subtract(t1).shiftRight(1);
        t1 = t1.subtract(tm1).subtract(vinf);
        t2 = t2.subtract(vinf.shiftLeft(1));
        tm1 = tm1.subtract(t2);

        // Number of bits to shift left.
        int ss = k*32;

        return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
    }

    // Division

    
    BigInteger divide(BigInteger val) {
        if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
                mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
            return divideKnuth(val);
        } else {
            return divideBurnikelZiegler(val);
        }
    }

    
    BigInteger divideKnuth(BigInteger val) {
        MutableBigInteger q = MutableBigInteger(),
                          a = MutableBigInteger(this->mag),
                          b = MutableBigInteger(val.mag);

        a.divideKnuth(b, q, false);
        return q.toBigInteger(this->signum * val.signum);
    }

    
    std::vector<BigInteger> divideAndRemainder(BigInteger val) {
        if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
                mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
            return divideAndRemainderKnuth(val);
        } else {
            return divideAndRemainderBurnikelZiegler(val);
        }
    }

    
    std::vector<BigInteger> divideAndRemainderKnuth(BigInteger val) {
        std::vector<BigInteger> result();
        MutableBigInteger q = MutableBigInteger(),
                          a = MutableBigInteger(this->mag),
                          b = MutableBigInteger(val.mag);
        MutableBigInteger r = a.divideKnuth(b, q);
        result[0] = q.toBigInteger(this->signum == val.signum ? 1 : -1);
        result[1] = r.toBigInteger(this->signum);
        return result;
    }

    
    BigInteger remainder(BigInteger val) {
        if (val.mag.size() < BURNIKEL_ZIEGLER_THRESHOLD ||
                mag.size() - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
            return remainderKnuth(val);
        } else {
            return remainderBurnikelZiegler(val);
        }
    }

    
    BigInteger remainderKnuth(BigInteger val) {
        MutableBigInteger q = MutableBigInteger(),
                          a = MutableBigInteger(this->mag),
                          b = MutableBigInteger(val.mag);

        return a.divideKnuth(b, q).toBigInteger(this->signum);
    }

    
    BigInteger divideBurnikelZiegler(BigInteger val) {
        return divideAndRemainderBurnikelZiegler(val)[0];
    }

    
    BigInteger remainderBurnikelZiegler(BigInteger val) {
        return divideAndRemainderBurnikelZiegler(val)[1];
    }

    
    std::vector<BigInteger> divideAndRemainderBurnikelZiegler(BigInteger val) {
        MutableBigInteger q = MutableBigInteger();
        MutableBigInteger r = MutableBigInteger(*this).divideAndRemainderBurnikelZiegler(MutableBigInteger(val), q);
        BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
        BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
        return std::vector<BigInteger> = {qBigInt, rBigInt};
    }

    
    BigInteger pow(int exponent) {
        assert(exponent > 0);
		if (signum == 0) {
            return (exponent == 0 ? ONE : this);
        }

        BigInteger partToSquare = this->abs();

        // Factor out powers of two from the base, as the exponentiation of
        // these can be done by left shifts only.
        // The remaining part can then be exponentiated faster.  The
        // powers of two will be multiplied back at the end.
        int powersOfTwo = partToSquare.getLowestSetBit();
        long bitsToShift = (std::int64_t)powersOfTwo * exponent;
        if (bitsToShift > std::numeric_limits<int>::max()) {
            reportOverflow();
        }

        int remainingBits;

        // Factor the powers of two out quickly by shifting right, if needed.
        if (powersOfTwo > 0) {
            partToSquare = partToSquare.shiftRight(powersOfTwo);
            remainingBits = partToSquare.bitLength();
            if (remainingBits == 1) {  // Nothing left but +/- 1?
                if (signum < 0 && (exponent&1) == 1) {
                    return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
                } else {
                    return ONE.shiftLeft(powersOfTwo*exponent);
                }
            }
        } else {
            remainingBits = partToSquare.bitLength();
            if (remainingBits == 1) { // Nothing left but +/- 1?
                if (signum < 0  && (exponent&1) == 1) {
                    return NEGATIVE_ONE;
                } else {
                    return ONE;
                }
            }
        }

        // This is a quick way to approximate the size of the result,
        // similar to doing log2[n] * exponent.  This will give an upper bound
        // of how big the result can be, and which algorithm to use.
        long scaleFactor = (long)remainingBits * exponent;

        // Use slightly different algorithms for small and large operands.
        // See if the result will safely fit into a long. (Largest 2^63-1)
        if (partToSquare.mag.size() == 1 && scaleFactor <= 62) {
            // Small number algorithm.  Everything fits into a long.
            int newSign = (signum <0  && (exponent&1) == 1 ? -1 : 1);
            long result = 1;
            long baseToPow2 = partToSquare.mag[0] & LONG_MASK;

            int workingExponent = exponent;

            // Perform exponentiation using repeated squaring trick
            while (workingExponent != 0) {
                if ((workingExponent & 1) == 1) {
                    result = result * baseToPow2;
                }

                if ((workingExponent >>>= 1) != 0) {
                    baseToPow2 = baseToPow2 * baseToPow2;
                }
            }

            // Multiply back the powers of two (quickly, by shifting left)
            if (powersOfTwo > 0) {
                if (bitsToShift + scaleFactor <= 62) { // Fits in long?
                    return valueOf((result << bitsToShift) * newSign);
                } else {
                    return valueOf(result*newSign).shiftLeft((int) bitsToShift);
                }
            }
            else {
                return valueOf(result*newSign);
            }
        } else {
            // Large number algorithm.  This is basically identical to
            // the algorithm above, but calls multiply() and square()
            // which may use more efficient algorithms for large numbers.
            BigInteger answer = ONE;

            int workingExponent = exponent;
            // Perform exponentiation using repeated squaring trick
            while (workingExponent != 0) {
                if ((workingExponent & 1) == 1) {
                    answer = answer.multiply(partToSquare);
                }

                if ((workingExponent >>>= 1) != 0) {
                    partToSquare = partToSquare.square();
                }
            }
            // Multiply back the (exponentiated) powers of two (quickly,
            // by shifting left)
            if (powersOfTwo > 0) {
                answer = answer.shiftLeft(powersOfTwo*exponent);
            }

            if (signum < 0 && (exponent&1) == 1) {
                return answer.negate();
            } else {
                return answer;
            }
        }
    }

    
    BigInteger gcd(BigInteger val) {
        if (val.signum == 0)
            return this->abs();
        else if (this.signum == 0)
            return val.abs();

        MutableBigInteger a = MutableBigInteger(this);
        MutableBigInteger b = MutableBigInteger(val);

        MutableBigInteger result = a.hybridGCD(b);

        return result.toBigInteger(1);
    }

    
    static int bitLengthForInt(int n) {
        return 32 - Integer.numberOfLeadingZeros(n);
    }

    
    static std::vector<int> leftShift(std::vector<int> a, int len, unsigned 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)) {
                std::vector<int> result(nInts + len);
                std::copy(a.begin(),a.begin() + len, result.begin());
                primitiveLeftShift(result, result.length, nBits);
                return result;
            } else {
                std::vector<int> resul(nInts + len + 1);
                std::copy(a.begin(), a.begin() + len, result.begin());
                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(std::vector<int> a, int len, int n) {
        int n2 = 32 - n;
        for (int i=len-1, c=a[i]; i > 0; i--) {
            unsigned int b = c;
            c = a[i-1];
            a[i] = (c << n2) | (b >> n);
        }
        a[0] = ((unsigned int)a[0]) >> n;
    }

    // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
    static void primitiveLeftShift(std::vector<int> a, int len, int n) {
        if (len == 0 || n == 0)
            return;

        int n2 = 32 - n;
        for (unsigned 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;
    }

    
    static int bitLength(std::vector<int> val, int len) {
        if (len == 0)
            return 0;
        return ((len - 1) << 5) + bitLengthForInt(val[0]);
    }

    
    BigInteger abs() {
        return (signum >= 0 ? this : this->negate());
    }

    
    BigInteger negate() {
        return BigInteger(this->mag, -this->signum);
    }

    
    int signum() {
        return this->signum;
    }

    // Modular Arithmetic Operations

    
    BigInteger mod(BigInteger m) {
        assert(m.signum > 0)

        BigInteger result = this->remainder(m);
        return (result.signum >= 0 ? result : result.add(m));
    }

    
    BigInteger modPow(BigInteger exponent, BigInteger m) {
        assert(m.signum > 0)

        // 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);

            if (m.mag.length < MAX_MAG_LENGTH / 2) {
                result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m);
            } else {
                MutableBigInteger t1;
                new MutableBigInteger(a1.multiply(m2)).multiply(MutableBigInteger(y1), t1);
                MutableBigInteger t2;
                new MutableBigInteger(a2.multiply(m1)).multiply(MutableBigInteger(y2), t2);
                t1.add(t2);
                MutableBigInteger q;
                result = t1.divide(MutableBigInteger(m), q).toBigInteger();
            }
        }

        return (invertResult ? result.modInverse(m) : result);
    }

    // Montgomery multiplication.  These are wrappers for
    // implMontgomeryXX routines which are expected to be replaced by
    // virtual machine intrinsics.  We don't use the intrinsics for
    // very large operands: MONTGOMERY_INTRINSIC_THRESHOLD should be
    // larger than any reasonable crypto key.
    static std::vector<int> montgomeryMultiply(std::vector<int> a, std::vector<int> b, std::vector<int> n, int len, long inv,
                                            std::vector<int> product) {
        implMontgomeryMultiplyChecks(a, b, n, len, product);
        if (len > MONTGOMERY_INTRINSIC_THRESHOLD) {
            // Very long argument: do not use an intrinsic
            product = multiplyToLen(a, len, b, len, product);
            return montReduce(product, n, len, (int)inv);
        } else {
            return implMontgomeryMultiply(a, b, n, len, inv, materialize(product, len));
        }
    }
    static std::vector<int> montgomerySquare(std::vector<int> a, std::vector<int> n, int len, std::int64_t inv,
                                          std::vector<int> product) {
        implMontgomeryMultiplyChecks(a, a, n, len, product);
        if (len > MONTGOMERY_INTRINSIC_THRESHOLD) {
            // Very long argument: do not use an intrinsic
            product = squareToLen(a, len, product);
            return montReduce(product, n, len, (int)inv);
        } else {
            return implMontgomerySquare(a, n, len, inv, materialize(product, len));
        }
    }

    // Range-check everything.
    static void implMontgomeryMultiplyChecks
        (std::vector<int> a, std::vector<int> b, std::vector<int> n, int len, std::vector<int> product) {
        if (len % 2 != 0) {
            throw std::invalid_argument("input array length must be even: " + std::to_string(len));
        }

        if (len < 1) {
            throw std::invalid_argument("invalid input length: " + std::to_string(len));
        }

        if (len > a.size() ||
            len > b.size() ||
            len > n.size() ||
            (len > product.size())) {
            throw std::invalid_argument("input array length out of bound: " + len);
        }
    }

    // Make sure that the int array z (which is expected to contain
    // the result of a Montgomery multiplication) is present and
    // sufficiently large.
    static std::vector<int> materialize(std::vector<int> z, int len) {
         if (z.size() < len)
             z = std::vector<int>(len);
         return z;
    }

    // These methods are intended to be be replaced by virtual machine
    // intrinsics.
    static std::vector<int> implMontgomeryMultiply(std::vector<int> a, std::vector<int> b, std::vector<int> n, int len,
                                         std::int64_t inv, std::vector<int> product) {
        product = multiplyToLen(a, len, b, len, product);
        return montReduce(product, n, len, (int)inv);
    }
    static std::vector<int> implMontgomerySquare(std::vector<int> a, std::vector<int> n, int len,
                                       std::int64_t inv, std::vector<int> product) {
        product = squareToLen(a, len, product);
        return montReduce(product, n, len, (int)inv);
    }

    static std::vector<int> bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
                                                std::numeric_limits<int>::max()}; // Sentinel

    
    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;

        std::vector<int> base = mag.clone();
        std::vector<int> exp = y.mag;
        std::vector<int> mod = z.mag;
        int modLen = mod.size();

        // Make modLen even. It is conventional to use a cryptographic
        // modulus that is 512, 768, 1024, or 2048 bits, so this code
        // will not normally be executed. However, it is necessary for
        // the correct functioning of the HotSpot intrinsics.
        if ((modLen & 1) != 0) {
            std::vector<int> x(modlen + 1);
            std::copy(mod.begin(), mod.begin() + modLen, x.begin());
            mod = std::move(x);
            modLen++;
        }

        // Select an appropriate window size
        int wbits = 0;
        int ebits = bitLength(exp, exp.size());
        // 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
        std::vector<std::vector<int>> table(tblmask);// = new int[tblmask][];
        for (int i=0; i < tblmask; i++)
            table[i] = std::vector<int>(modLen);

        // Compute the modular inverse of the least significant 64-bit
        // digit of the modulus
        std::int64_t n0 = (mod[modLen-1] & LONG_MASK) + ((mod[modLen-2] & LONG_MASK) << 32);
        std::int64_t inv = -MutableBigInteger.inverseMod64(n0);

        // Convert base to Montgomery form
        std::vector<int> a = leftShift(base, base.length, modLen << 5);

        MutableBigInteger q = MutableBigInteger(),
                          a2 = MutableBigInteger(a),
                          b2 = MutableBigInteger(mod);
        b2.normalize(); // MutableBigInteger.divide() assumes that its
                        // divisor is in normal form.

        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].size() < modLen) {
           int offset = modLen - table[0].size();
           std::vector<int> t2(modLen);// = new int[modLen];
           std::copy(table[0].begin(),table[0].end(), t2.begin());
           table[0] = t2;
        }

        // Set b to the square of the base
        std::vector<int> b = montgomerySquare(table[0], mod, modLen, inv, null);

        // Set t to high half of b
        std::vector<int> t(b.begin(), b.begin() + modLen);

        // Fill in the table with odd powers of the base
        for (int i=1; i < tblmask; i++) {
            table[i] = montgomeryMultiply(t, table[i-1], mod, modLen, inv, null);
        }

        // Pre load the window that slides over the exponent
        unsigned 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 = montgomeryMultiply(t, mult, mod, modLen, inv, a);
                    t = a; a = b; b = t;
                }
            }

            // Check if done
            if (ebits == 0)
                break;

            // Square the input
            if (!isone) {
                t = b;
                a = montgomerySquare(t, mod, modLen, inv, a);
                t = a; a = b; b = t;
            }
        }

        // Convert result out of Montgomery form and return
        int[] t2 = new int[2*modLen];
        System.arraycopy(b, 0, t2, modLen, modLen);

        b = montReduce(t2, mod, modLen, (int)inv);

        t2 = Arrays.copyOf(b, modLen);

        return new BigInteger(1, t2);
    }

    
    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.
     */
    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;
    }

    
    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);
    }

    
    static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
        implMulAddCheck(out, in, offset, len, k);
        return implMulAdd(out, in, offset, len, k);
    }

    
    static void implMulAddCheck(int[] out, int[] in, int offset, int len, int k) {
        if (len > in.length) {
            throw new IllegalArgumentException("input length is out of bound: " + len + " > " + in.length);
        }
        if (offset < 0) {
            throw new IllegalArgumentException("input offset is invalid: " + offset);
        }
        if (offset > (out.length - 1)) {
            throw new IllegalArgumentException("input offset is out of bound: " + offset + " > " + (out.length - 1));
        }
        if (len > (out.length - offset)) {
            throw new IllegalArgumentException("input len is out of bound: " + len + " > " + (out.length - offset));
        }
    }

    
    static int implMulAdd(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;
    }

    
    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;
    }

    
    BigInteger modPow2(BigInteger exponent, int p) {
        /*
         * Perform exponentiation using repeated squaring trick, chopping off
         * high order bits as indicated by modulus.
         */
        BigInteger result = ONE;
        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;
    }

    
    BigInteger mod2(int p) {
        if (bitLength() <= p)
            return this;

        // Copy remaining ints of mag
        int numInts = (p + 31) >>> 5;
        int[] mag = new int[numInts];
        System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, 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));
    }

    
    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

    
    BigInteger shiftLeft(int n) {
        if (signum == 0)
            return ZERO;
        if (n > 0) {
            return new BigInteger(shiftLeft(mag, n), signum);
        } else if (n == 0) {
            return this;
        } else {
            // Possible int overflow in (-n) is not a trouble,
            // because shiftRightImpl considers its argument unsigned
            return shiftRightImpl(-n);
        }
    }

    
    static int[] shiftLeft(int[] mag, int n) {
        int nInts = n >>> 5;
        int nBits = n & 0x1f;
        int magLen = mag.length;
        int newMag[] = null;

        if (nBits == 0) {
            newMag = new int[magLen + nInts];
            System.arraycopy(mag, 0, newMag, 0, magLen);
        } 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 newMag;
    }

    
    BigInteger shiftRight(int n) {
        if (signum == 0)
            return ZERO;
        if (n > 0) {
            return shiftRightImpl(n);
        } else if (n == 0) {
            return this;
        } else {
            return BigInteger(shiftLeft(mag, -n), signum);
        }
    }

    
    BigInteger shiftRightImpl(int 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 = Arrays.copyOf(mag, newMagLen);
        } 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

    
    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);
    }

    
    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);
    }

    
    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);
    }

    
    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);
    }

    
    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

    
    boolean testBit(int n) {
        if (n < 0)
            throw new ArithmeticException("Negative bit address");

        return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
    }

    
    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);
    }

    
    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);
    }

    
    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);
    }

    
    int getLowestSetBit() {
        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

    
    int bitLength() {
        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;
    }

    
    int bitCount() {
        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

    
    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

    
    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;
    }

    
    const 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;
    }

    
    const int compareMagnitude(long val) {
        assert val != Long.MIN_VALUE;
        int[] m1 = mag;
        int len = m1.length;
        if (len > 2) {
            return 1;
        }
        if (val < 0) {
            val = -val;
        }
        int highWord = (int)(val >>> 32);
        if (highWord == 0) {
            if (len < 1)
                return -1;
            if (len > 1)
                return 1;
            int a = m1[0];
            int b = (int)val;
            if (a != b) {
                return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
            }
            return 0;
        } else {
            if (len < 2)
                return -1;
            int a = m1[0];
            int b = highWord;
            if (a != b) {
                return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
            }
            a = m1[1];
            b = (int)val;
            if (a != b) {
                return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
            }
            return 0;
        }
    }

    
    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;
    }

    
    BigInteger min(BigInteger val) {
        return (compareTo(val) < 0 ? this : val);
    }

    
    BigInteger max(BigInteger val) {
        return (compareTo(val) > 0 ? this : val);
    }


    // Hash Function

    
    int hashCode() {
        int hashCode = 0;

        for (int i=0; i < mag.length; i++)
            hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));

        return hashCode * signum;
    }

    
    String toString(int radix) {
        if (signum == 0)
            return "0";
        if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
            radix = 10;

        // If it's small enough, use smallToString.
        if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
           return smallToString(radix);

        // Otherwise use recursive toString, which requires positive arguments.
        // The results will be concatenated into this StringBuilder
        StringBuilder sb = new StringBuilder();
        if (signum < 0) {
            toString(this.negate(), sb, radix, 0);
            sb.insert(0, '-');
        }
        else
            toString(this, sb, radix, 0);

        return sb.toString();
    }

    
    String smallToString(int radix) {
        if (signum == 0) {
            return "0";
        }

        // Compute upper bound on number of digit groups and allocate space
        int maxNumDigitGroups = (4*mag.length + 6)/7;
        String digitGroup[] = new String[maxNumDigitGroups];

        // Translate number to string, a digit group at a time
        BigInteger tmp = this.abs();
        int numGroups = 0;
        while (tmp.signum != 0) {
            BigInteger d = longRadix[radix];

            MutableBigInteger q = new MutableBigInteger(),
                              a = new MutableBigInteger(tmp.mag),
                              b = new MutableBigInteger(d.mag);
            MutableBigInteger r = a.divide(b, q);
            BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
            BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);

            digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
            tmp = q2;
        }

        // Put sign (if any) and first digit group into result buffer
        StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
        if (signum < 0) {
            buf.append('-');
        }
        buf.append(digitGroup[numGroups-1]);

        // Append remaining digit groups padded with leading zeros
        for (int i=numGroups-2; i >= 0; i--) {
            // Prepend (any) leading zeros for this digit group
            int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
            if (numLeadingZeros != 0) {
                buf.append(zeros[numLeadingZeros]);
            }
            buf.append(digitGroup[i]);
        }
        return buf.toString();
    }

    
    static void toString(BigInteger u, StringBuilder sb, int radix,
                                 int digits) {
        /* If we're smaller than a certain threshold, use the smallToString
           method, padding with leading zeroes when necessary. */
        if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
            String s = u.smallToString(radix);

            // Pad with internal zeros if necessary.
            // Don't pad if we're at the beginning of the string.
            if ((s.length() < digits) && (sb.length() > 0)) {
                for (int i=s.length(); i < digits; i++) { // May be a faster way to
                    sb.append('0');                    // do this?
                }
            }

            sb.append(s);
            return;
        }

        int b, n;
        b = u.bitLength();

        // Calculate a value for n in the equation radix^(2^n) = u
        // and subtract 1 from that value.  This is used to find the
        // cache index that contains the best value to divide u.
        n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
        BigInteger v = getRadixConversionCache(radix, n);
        BigInteger[] results;
        results = u.divideAndRemainder(v);

        int expectedDigits = 1 << n;

        // Now recursively build the two halves of each number.
        toString(results[0], sb, radix, digits-expectedDigits);
        toString(results[1], sb, radix, expectedDigits);
    }

    
    static BigInteger getRadixConversionCache(int radix, int exponent) {
        BigInteger[] cacheLine = powerCache[radix]; // volatile read
        if (exponent < cacheLine.length) {
            return cacheLine[exponent];
        }

        int oldLength = cacheLine.length;
        cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
        for (int i = oldLength; i <= exponent; i++) {
            cacheLine[i] = cacheLine[i - 1].pow(2);
        }

        BigInteger[][] pc = powerCache; // volatile read again
        if (exponent >= pc[radix].length) {
            pc = pc.clone();
            pc[radix] = cacheLine;
            powerCache = pc; // volatile write, publish
        }
        return cacheLine[exponent];
    }

    /* zero[i] is a string of i consecutive zeros. */
    static String zeros[] = new String[64];
    static {
        zeros[63] =
            "000000000000000000000000000000000000000000000000000000000000000";
        for (int i=0; i < 63; i++)
            zeros[i] = zeros[63].substring(0, i);
    }

    
    String toString() {
        return toString(10);
    }

    
    byte[] toByteArray() {
        int byteLen = bitLength()/8 + 1;
        byte[] byteArray = new byte[byteLen];

        for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i >= 0; i--) {
            if (bytesCopied == 4) {
                nextInt = getInt(intIndex++);
                bytesCopied = 1;
            } else {
                nextInt >>>= 8;
                bytesCopied++;
            }
            byteArray[i] = (byte)nextInt;
        }
        return byteArray;
    }

    
    int intValue() {
        int result = 0;
        result = getInt(0);
        return result;
    }

    
    long longValue() {
        long result = 0;

        for (int i=1; i >= 0; i--)
            result = (result << 32) + (getInt(i) & LONG_MASK);
        return result;
    }

    
    float floatValue() {
        if (signum == 0) {
            return 0.0f;
        }

        int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;

        // exponent == floor(log2(abs(this)))
        if (exponent < Long.SIZE - 1) {
            return longValue();
        } else if (exponent > Float.MAX_EXPONENT) {
            return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
        }

        /*
         * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
         * one bit. To make rounding easier, we pick out the top
         * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
         * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
         * bits, and signifFloor the top SIGNIFICAND_WIDTH.
         *
         * It helps to consider the real number signif = abs(this) *
         * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
         */
        int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;

        int twiceSignifFloor;
        // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
        // We do the shift into an int directly to improve performance.

        int nBits = shift & 0x1f;
        int nBits2 = 32 - nBits;

        if (nBits == 0) {
            twiceSignifFloor = mag[0];
        } else {
            twiceSignifFloor = mag[0] >>> nBits;
            if (twiceSignifFloor == 0) {
                twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
            }
        }

        int signifFloor = twiceSignifFloor >> 1;
        signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit

        /*
         * We round up if either the fractional part of signif is strictly
         * greater than 0.5 (which is true if the 0.5 bit is set and any lower
         * bit is set), or if the fractional part of signif is >= 0.5 and
         * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
         * are set). This is equivalent to the desired HALF_EVEN rounding.
         */
        boolean increment = (twiceSignifFloor & 1) != 0
                && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
        int signifRounded = increment ? signifFloor + 1 : signifFloor;
        int bits = ((exponent + FloatConsts.EXP_BIAS))
                << (FloatConsts.SIGNIFICAND_WIDTH - 1);
        bits += signifRounded;
        /*
         * If signifRounded == 2^24, we'd need to set all of the significand
         * bits to zero and add 1 to the exponent. This is exactly the behavior
         * we get from just adding signifRounded to bits directly. If the
         * exponent is Float.MAX_EXPONENT, we round up (correctly) to
         * Float.POSITIVE_INFINITY.
         */
        bits |= signum & FloatConsts.SIGN_BIT_MASK;
        return Float.intBitsToFloat(bits);
    }

    
    double doubleValue() {
        if (signum == 0) {
            return 0.0;
        }

        int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;

        // exponent == floor(log2(abs(this))Double)
        if (exponent < Long.SIZE - 1) {
            return longValue();
        } else if (exponent > Double.MAX_EXPONENT) {
            return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
        }

        /*
         * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
         * one bit. To make rounding easier, we pick out the top
         * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
         * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
         * bits, and signifFloor the top SIGNIFICAND_WIDTH.
         *
         * It helps to consider the real number signif = abs(this) *
         * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
         */
        int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;

        long twiceSignifFloor;
        // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
        // We do the shift into a long directly to improve performance.

        int nBits = shift & 0x1f;
        int nBits2 = 32 - nBits;

        int highBits;
        int lowBits;
        if (nBits == 0) {
            highBits = mag[0];
            lowBits = mag[1];
        } else {
            highBits = mag[0] >>> nBits;
            lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
            if (highBits == 0) {
                highBits = lowBits;
                lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
            }
        }

        twiceSignifFloor = ((highBits & LONG_MASK) << 32)
                | (lowBits & LONG_MASK);

        long signifFloor = twiceSignifFloor >> 1;
        signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit

        /*
         * We round up if either the fractional part of signif is strictly
         * greater than 0.5 (which is true if the 0.5 bit is set and any lower
         * bit is set), or if the fractional part of signif is >= 0.5 and
         * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
         * are set). This is equivalent to the desired HALF_EVEN rounding.
         */
        boolean increment = (twiceSignifFloor & 1) != 0
                && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
        long signifRounded = increment ? signifFloor + 1 : signifFloor;
        long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
                << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
        bits += signifRounded;
        /*
         * If signifRounded == 2^53, we'd need to set all of the significand
         * bits to zero and add 1 to the exponent. This is exactly the behavior
         * we get from just adding signifRounded to bits directly. If the
         * exponent is Double.MAX_EXPONENT, we round up (correctly) to
         * Double.POSITIVE_INFINITY.
         */
        bits |= signum & DoubleConsts.SIGN_BIT_MASK;
        return Double.longBitsToDouble(bits);
    }

    
    static int[] stripLeadingZeroInts(int val[]) {
        int vlen = val.length;
        int keep;

        // Find first nonzero byte
        for (keep = 0; keep < vlen && val[keep] == 0; keep++)
            ;
        return java.util.Arrays.copyOfRange(val, keep, vlen);
    }

    
    static int[] trustedStripLeadingZeroInts(int val[]) {
        int vlen = val.length;
        int keep;

        // Find first nonzero byte
        for (keep = 0; keep < vlen && val[keep] == 0; keep++)
            ;
        return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
    }

    
    static int[] stripLeadingZeroBytes(byte a[]) {
        int byteLength = a.length;
        int keep;

        // Find first nonzero byte
        for (keep = 0; keep < byteLength && a[keep] == 0; keep++)
            ;

        // Allocate new array and copy relevant part of input array
        int intLength = ((byteLength - keep) + 3) >>> 2;
        int[] result = new int[intLength];
        int b = byteLength - 1;
        for (int i = intLength-1; i >= 0; i--) {
            result[i] = a[b--] & 0xff;
            int bytesRemaining = b - keep + 1;
            int bytesToTransfer = Math.min(3, bytesRemaining);
            for (int j=8; j <= (bytesToTransfer << 3); j += 8)
                result[i] |= ((a[b--] & 0xff) << j);
        }
        return result;
    }

    
    static int[] makePositive(byte a[]) {
        int keep, k;
        int byteLength = a.length;

        // Find first non-sign (0xff) byte of input
        for (keep=0; keep < byteLength && a[keep] == -1; keep++)
            ;


        /* Allocate output array.  If all non-sign bytes are 0x00, we must
         * allocate space for one extra output byte. */
        for (k=keep; k < byteLength && a[k] == 0; k++)
            ;

        int extraByte = (k == byteLength) ? 1 : 0;
        int intLength = ((byteLength - keep + extraByte) + 3) >>> 2;
        int result[] = new int[intLength];

        /* Copy one's complement of input into output, leaving extra
         * byte (if it exists) == 0x00 */
        int b = byteLength - 1;
        for (int i = intLength-1; i >= 0; i--) {
            result[i] = a[b--] & 0xff;
            int numBytesToTransfer = Math.min(3, b-keep+1);
            if (numBytesToTransfer < 0)
                numBytesToTransfer = 0;
            for (int j=8; j <= 8*numBytesToTransfer; j += 8)
                result[i] |= ((a[b--] & 0xff) << j);

            // Mask indicates which bits must be complemented
            int mask = -1 >>> (8*(3-numBytesToTransfer));
            result[i] = ~result[i] & mask;
        }

        // Add one to one's complement to generate two's complement
        for (int i=result.length-1; i >= 0; i--) {
            result[i] = (int)((result[i] & LONG_MASK) + 1);
            if (result[i] != 0)
                break;
        }

        return result;
    }

    
    static int[] makePositive(int a[]) {
        int keep, j;

        // Find first non-sign (0xffffffff) int of input
        for (keep=0; keep < a.length && a[keep] == -1; keep++)
            ;

        /* Allocate output array.  If all non-sign ints are 0x00, we must
         * allocate space for one extra output int. */
        for (j=keep; j < a.length && a[j] == 0; j++)
            ;
        int extraInt = (j == a.length ? 1 : 0);
        int result[] = new int[a.length - keep + extraInt];

        /* Copy one's complement of input into output, leaving extra
         * int (if it exists) == 0x00 */
        for (int i = keep; i < a.length; i++)
            result[i - keep + extraInt] = ~a[i];

        // Add one to one's complement to generate two's complement
        for (int i=result.length-1; ++result[i] == 0; i--)
            ;

        return result;
    }

    /*
     * The following two arrays are used for fast String conversions.  Both
     * are indexed by radix.  The first is the number of digits of the given
     * radix that can fit in a Java long without "going negative", i.e., the
     * highest integer n such that radix**n < 2**63.  The second is the
     * "long radix" that tears each number into "long digits", each of which
     * consists of the number of digits in the corresponding element in
     * digitsPerLong (longRadix[i] = i**digitPerLong[i]).  Both arrays have
     * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
     * used.
     */
    static int digitsPerLong[] = {0, 0,
        62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
        14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};

    static BigInteger longRadix[] = {null, null,
        valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
        valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
        valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
        valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
        valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
        valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
        valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
        valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
        valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
        valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
        valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
        valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
        valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
        valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
        valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
        valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
        valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
        valueOf(0x41c21cb8e1000000L)};

    /*
     * These two arrays are the integer analogue of above.
     */
    static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
        11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
        6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};

    static int intRadix[] = {0, 0,
        0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
        0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
        0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f,  0x10000000,
        0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
        0x6c20a40,  0x8d2d931,  0xb640000,  0xe8d4a51,  0x1269ae40,
        0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
        0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
    };

    

    
    int intLength() {
        return (bitLength() >>> 5) + 1;
    }

    /* Returns sign bit */
    int signBit() {
        return signum < 0 ? 1 : 0;
    }

    /* Returns an int of sign bits */
    int signInt() {
        return signum < 0 ? -1 : 0;
    }

    
    int getInt(int n) {
        if (n < 0)
            return 0;
        if (n >= mag.length)
            return signInt();

        int magInt = mag[mag.length-n-1];

        return (signum >= 0 ? magInt :
                (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
    }

    
    int firstNonzeroIntNum() {
        int fn = firstNonzeroIntNum - 2;
        if (fn == -2) { // firstNonzeroIntNum not initialized yet
            fn = 0;

            // Search for the first nonzero int
            int i;
            int mlen = mag.length;
            for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
                ;
            fn = mlen - i - 1;
            firstNonzeroIntNum = fn + 2; // offset by two to initialize
        }
        return fn;
    }

    
    static const long serialVersionUID = -8287574255936472291L;

    
    static const ObjectStreamField[] serialPersistentFields = {
        new ObjectStreamField("signum", Integer.TYPE),
        new ObjectStreamField("magnitude", byte[].class),
        new ObjectStreamField("bitCount", Integer.TYPE),
        new ObjectStreamField("bitLength", Integer.TYPE),
        new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
        new ObjectStreamField("lowestSetBit", Integer.TYPE)
        };

    
    void readObject(java.io.ObjectInputStream s)
        throws java.io.IOException, ClassNotFoundException {
        /*
         * In order to maintain compatibility with previous serialized forms,
         * the magnitude of a BigInteger is serialized as an array of bytes.
         * The magnitude field is used as a temporary store for the byte array
         * that is deserialized. The cached computation fields should be
         * transient but are serialized for compatibility reasons.
         */

        // prepare to read the alternate persistent fields
        ObjectInputStream.GetField fields = s.readFields();

        // Read the alternate persistent fields that we care about
        int sign = fields.get("signum", -2);
        byte[] magnitude = (byte[])fields.get("magnitude", null);

        // Validate signum
        if (sign < -1 || sign > 1) {
            String message = "BigInteger: Invalid signum value";
            if (fields.defaulted("signum"))
                message = "BigInteger: Signum not present in stream";
            throw new java.io.StreamCorruptedException(message);
        }
        int[] mag = stripLeadingZeroBytes(magnitude);
        if ((mag.length == 0) != (sign == 0)) {
            String message = "BigInteger: signum-magnitude mismatch";
            if (fields.defaulted("magnitude"))
                message = "BigInteger: Magnitude not present in stream";
            throw new java.io.StreamCorruptedException(message);
        }

        // Commit const fields via Unsafe
        UnsafeHolder.putSign(this, sign);

        // Calculate mag field from magnitude and discard magnitude
        UnsafeHolder.putMag(this, mag);
        if (mag.length >= MAX_MAG_LENGTH) {
            try {
                checkRange();
            } catch (ArithmeticException e) {
                throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range");
            }
        }
    }

    // Support for resetting const fields while deserializing
    static class UnsafeHolder {
        static const sun.misc.Unsafe unsafe;
        static const long signumOffset;
        static const long magOffset;
        static {
            try {
                unsafe = sun.misc.Unsafe.getUnsafe();
                signumOffset = unsafe.objectFieldOffset
                    (BigInteger.class.getDeclaredField("signum"));
                magOffset = unsafe.objectFieldOffset
                    (BigInteger.class.getDeclaredField("mag"));
            } catch (Exception ex) {
                throw new ExceptionInInitializerError(ex);
            }
        }

        static void putSign(BigInteger bi, int sign) {
            unsafe.putIntVolatile(bi, signumOffset, sign);
        }

        static void putMag(BigInteger bi, int[] magnitude) {
            unsafe.putObjectVolatile(bi, magOffset, magnitude);
        }
    }

    
    void writeObject(ObjectOutputStream s) throws IOException {
        // set the values of the Serializable fields
        ObjectOutputStream.PutField fields = s.putFields();
        fields.put("signum", signum);
        fields.put("magnitude", magSerializedForm());
        // The values written for cached fields are compatible with older
        // versions, but are ignored in readObject so don't otherwise matter.
        fields.put("bitCount", -1);
        fields.put("bitLength", -1);
        fields.put("lowestSetBit", -2);
        fields.put("firstNonzeroByteNum", -2);

        // save them
        s.writeFields();
}

    
    byte[] magSerializedForm() {
        int len = mag.length;

        int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
        int byteLen = (bitLen + 7) >>> 3;
        byte[] result = new byte[byteLen];

        for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
             i >= 0; i--) {
            if (bytesCopied == 4) {
                nextInt = mag[intIndex--];
                bytesCopied = 1;
            } else {
                nextInt >>>= 8;
                bytesCopied++;
            }
            result[i] = (byte)nextInt;
        }
        return result;
    }

    
    long longValueExact() {
        if (mag.length <= 2 && bitLength() <= 63)
            return longValue();
        else
            throw new ArithmeticException("BigInteger out of long range");
    }

    
    int intValueExact() {
        if (mag.length <= 1 && bitLength() <= 31)
            return intValue();
        else
            throw new ArithmeticException("BigInteger out of int range");
    }

    
    short shortValueExact() {
        if (mag.length <= 1 && bitLength() <= 31) {
            int value = intValue();
            if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
                return shortValue();
        }
        throw new ArithmeticException("BigInteger out of short range");
    }

    
    byte byteValueExact() {
        if (mag.length <= 1 && bitLength() <= 31) {
            int value = intValue();
            if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
                return byteValue();
        }
        throw new ArithmeticException("BigInteger out of byte range");
    }
    static void main(String[] args) {
    	BigInteger a = new BigInteger(6);
    	a = a.pow(100);
    	System.out.println(a);
    }
}