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Task/Pisano-period/Java/pisano-period.java
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263
Task/Pisano-period/Java/pisano-period.java
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import java.util.ArrayList;
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import java.util.Collections;
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import java.util.HashMap;
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import java.util.List;
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import java.util.Map;
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import java.util.TreeMap;
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public class PisanoPeriod {
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public static void main(String[] args) {
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System.out.printf("Print pisano(p^2) for every prime p lower than 15%n");
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for ( long i = 2 ; i < 15 ; i++ ) {
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if ( isPrime(i) ) {
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long n = i*i;
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System.out.printf("pisano(%d) = %d%n", n, pisano(n));
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}
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}
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System.out.printf("%nPrint pisano(p) for every prime p lower than 180%n");
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for ( long n = 2 ; n < 180 ; n++ ) {
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if ( isPrime(n) ) {
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System.out.printf("pisano(%d) = %d%n", n, pisano(n));
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}
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}
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System.out.printf("%nPrint pisano(n) for every integer from 1 to 180%n");
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for ( long n = 1 ; n <= 180 ; n++ ) {
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System.out.printf("%3d ", pisano(n));
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if ( n % 10 == 0 ) {
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System.out.printf("%n");
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}
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}
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}
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private static final boolean isPrime(long test) {
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if ( test == 2 ) {
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return true;
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}
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if ( test % 2 == 0 ) {
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return false;
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}
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for ( long i = 3 ; i <= Math.sqrt(test) ; i += 2 ) {
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if ( test % i == 0 ) {
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return false;
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}
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}
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return true;
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}
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private static Map<Long,Long> PERIOD_MEMO = new HashMap<>();
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static {
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PERIOD_MEMO.put(2L, 3L);
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PERIOD_MEMO.put(3L, 8L);
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PERIOD_MEMO.put(5L, 20L);
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}
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// See http://webspace.ship.edu/msrenault/fibonacci/fib.htm
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private static long pisano(long n) {
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if ( PERIOD_MEMO.containsKey(n) ) {
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return PERIOD_MEMO.get(n);
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}
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if ( n == 1 ) {
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return 1;
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}
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Map<Long,Long> factors = getFactors(n);
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// Special cases
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// pisano(2^k) = 3*n/2
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if ( factors.size() == 1 & factors.get(2L) != null && factors.get(2L) > 0 ) {
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long result = 3 * n / 2;
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PERIOD_MEMO.put(n, result);
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return result;
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}
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// pisano(5^k) = 4*n
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if ( factors.size() == 1 & factors.get(5L) != null && factors.get(5L) > 0 ) {
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long result = 4*n;
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PERIOD_MEMO.put(n, result);
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return result;
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}
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// pisano(2*5^k) = 6*n
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if ( factors.size() == 2 & factors.get(2L) != null && factors.get(2L) == 1 && factors.get(5L) != null && factors.get(5L) > 0 ) {
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long result = 6*n;
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PERIOD_MEMO.put(n, result);
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return result;
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}
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List<Long> primes = new ArrayList<>(factors.keySet());
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long prime = primes.get(0);
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if ( factors.size() == 1 && factors.get(prime) == 1 ) {
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List<Long> divisors = new ArrayList<>();
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if ( n % 10 == 1 || n % 10 == 9 ) {
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for ( long divisor : getDivisors(prime-1) ) {
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if ( divisor % 2 == 0 ) {
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divisors.add(divisor);
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}
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}
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}
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else {
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List<Long> pPlus1Divisors = getDivisors(prime+1);
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for ( long divisor : getDivisors(2*prime+2) ) {
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if ( ! pPlus1Divisors.contains(divisor) ) {
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divisors.add(divisor);
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}
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}
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}
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Collections.sort(divisors);
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for ( long divisor : divisors ) {
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if ( fibModIdentity(divisor, prime) ) {
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PERIOD_MEMO.put(prime, divisor);
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return divisor;
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}
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}
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throw new RuntimeException("ERROR 144: Divisor not found.");
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}
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long period = (long) Math.pow(prime, factors.get(prime)-1) * pisano(prime);
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for ( int i = 1 ; i < primes.size() ; i++ ) {
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prime = primes.get(i);
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period = lcm(period, (long) Math.pow(prime, factors.get(prime)-1) * pisano(prime));
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}
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PERIOD_MEMO.put(n, period);
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return period;
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}
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// Use Matrix multiplication to compute Fibonacci numbers.
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private static boolean fibModIdentity(long n, long mod) {
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long aRes = 0;
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long bRes = 1;
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long cRes = 1;
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long aBase = 0;
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long bBase = 1;
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long cBase = 1;
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while ( n > 0 ) {
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if ( n % 2 == 1 ) {
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long temp1 = 0, temp2 = 0, temp3 = 0;
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if ( aRes > SQRT || aBase > SQRT || bRes > SQRT || bBase > SQRT || cBase > SQRT || cRes > SQRT ) {
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temp1 = (multiply(aRes, aBase, mod) + multiply(bRes, bBase, mod)) % mod;
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temp2 = (multiply(aBase, bRes, mod) + multiply(bBase, cRes, mod)) % mod;
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temp3 = (multiply(bBase, bRes, mod) + multiply(cBase, cRes, mod)) % mod;
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}
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else {
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temp1 = ((aRes*aBase % mod) + (bRes*bBase % mod)) % mod;
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temp2 = ((aBase*bRes % mod) + (bBase*cRes % mod)) % mod;
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temp3 = ((bBase*bRes % mod) + (cBase*cRes % mod)) % mod;
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}
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aRes = temp1;
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bRes = temp2;
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cRes = temp3;
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}
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n >>= 1L;
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long temp1 = 0, temp2 = 0, temp3 = 0;
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if ( aBase > SQRT || bBase > SQRT || cBase > SQRT ) {
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temp1 = (multiply(aBase, aBase, mod) + multiply(bBase, bBase, mod)) % mod;
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temp2 = (multiply(aBase, bBase, mod) + multiply(bBase, cBase, mod)) % mod;
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temp3 = (multiply(bBase, bBase, mod) + multiply(cBase, cBase, mod)) % mod;
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}
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else {
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temp1 = ((aBase*aBase % mod) + (bBase*bBase % mod)) % mod;
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temp2 = ((aBase*bBase % mod) + (bBase*cBase % mod)) % mod;
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temp3 = ((bBase*bBase % mod) + (cBase*cBase % mod)) % mod;
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}
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aBase = temp1;
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bBase = temp2;
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cBase = temp3;
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}
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return aRes % mod == 0 && bRes % mod == 1 && cRes % mod == 1;
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}
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private static final long SQRT = (long) Math.sqrt(Long.MAX_VALUE);
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// Result is a*b % mod, without overflow.
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public static final long multiply(long a, long b, long modulus) {
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//System.out.println(" multiply : a = " + a + ", b = " + b + ", mod = " + modulus);
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long x = 0;
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long y = a % modulus;
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long t;
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while ( b > 0 ) {
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if ( b % 2 == 1 ) {
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t = x + y;
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x = (t > modulus ? t-modulus : t);
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}
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t = y << 1;
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y = (t > modulus ? t-modulus : t);
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b >>= 1;
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}
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//System.out.println(" multiply : answer = " + (x % modulus));
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return x % modulus;
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}
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private static final List<Long> getDivisors(long number) {
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List<Long> divisors = new ArrayList<>();
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long sqrt = (long) Math.sqrt(number);
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for ( long i = 1 ; i <= sqrt ; i++ ) {
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if ( number % i == 0 ) {
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divisors.add(i);
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long div = number / i;
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if ( div != i ) {
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divisors.add(div);
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}
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}
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}
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return divisors;
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}
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public static long lcm(long a, long b) {
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return a*b/gcd(a,b);
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}
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public static long gcd(long a, long b) {
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if ( b == 0 ) {
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return a;
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}
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return gcd(b, a%b);
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}
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private static final Map<Long,Map<Long,Long>> allFactors = new TreeMap<Long,Map<Long,Long>>();
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static {
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Map<Long,Long> factors = new TreeMap<Long,Long>();
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factors.put(2L, 1L);
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allFactors.put(2L, factors);
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}
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public static Long MAX_ALL_FACTORS = 100000L;
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public static final Map<Long,Long> getFactors(Long number) {
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if ( allFactors.containsKey(number) ) {
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return allFactors.get(number);
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}
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Map<Long,Long> factors = new TreeMap<Long,Long>();
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if ( number % 2 == 0 ) {
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Map<Long,Long> factorsdDivTwo = getFactors(number/2);
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factors.putAll(factorsdDivTwo);
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factors.merge(2L, 1L, (v1, v2) -> v1 + v2);
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if ( number < MAX_ALL_FACTORS ) {
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allFactors.put(number, factors);
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}
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return factors;
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}
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boolean prime = true;
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long sqrt = (long) Math.sqrt(number);
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for ( long i = 3 ; i <= sqrt ; i += 2 ) {
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if ( number % i == 0 ) {
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prime = false;
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factors.putAll(getFactors(number/i));
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factors.merge(i, 1L, (v1, v2) -> v1 + v2);
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if ( number < MAX_ALL_FACTORS ) {
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allFactors.put(number, factors);
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}
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return factors;
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}
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}
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if ( prime ) {
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factors.put(number, 1L);
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if ( number < MAX_ALL_FACTORS ) {
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allFactors.put(number, factors);
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}
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}
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return factors;
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}
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}
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