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# Java example source code file (SmallPrimes.java)

This example Java source code file (SmallPrimes.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

## Java - Java tags/keywords

arraylist, biginteger, list, math, primes_last, smallprimes, util

## The SmallPrimes.java Java example source code

```/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements.  See the NOTICE file distributed with
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License.  You may obtain a copy of the License at
*
*
* Unless required by applicable law or agreed to in writing, software
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
*/
package org.apache.commons.math3.primes;

import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;

import org.apache.commons.math3.util.FastMath;

/**
* Utility methods to work on primes within the <code>int range.
* @since 3.2
*/
class SmallPrimes {

/**
* The first 512 prime numbers.
* <p>
* It contains all primes smaller or equal to the cubic square of Integer.MAX_VALUE.
* As a result, <code>int numbers which are not reduced by those primes are guaranteed
* to be either prime or semi prime.
*/
public static final int[] PRIMES = {2,
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179,
181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283,
293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419,
421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547,
557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661,
673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811,
821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087,
1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229,
1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381,
1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523,
1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663,
1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823,
1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993,
1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131,
2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293,
2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437,
2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621,
2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749,
2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909,
2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083,
3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259,
3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433,
3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581,
3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671};

/** The last number in PRIMES. */
public static final int PRIMES_LAST = PRIMES[PRIMES.length - 1];

/**
* Hide utility class.
*/
private SmallPrimes() {
}

/**
* Extract small factors.
* @param n the number to factor, must be > 0.
* @param factors the list where to add the factors.
* @return the part of n which remains to be factored, it is either a prime or a semi-prime
*/
public static int smallTrialDivision(int n, final List<Integer> factors) {
for (int p : PRIMES) {
while (0 == n % p) {
n /= p;
}
}
return n;
}

/**
* Extract factors in the range <code>PRIME_LAST+2 to `maxFactors`.
* @param n the number to factorize, must be >= PRIME_LAST+2 and must not contain any factor below PRIME_LAST+2
* @param maxFactor the upper bound of trial division: if it is reached, the method gives up and returns n.
* @param factors the list where to add the factors.
* @return  n or 1 if factorization is completed.
*/
public static int boundedTrialDivision(int n, int maxFactor, List<Integer> factors) {
int f = PRIMES_LAST + 2;
// no check is done about n >= f
while (f <= maxFactor) {
if (0 == n % f) {
n /= f;
break;
}
f += 4;
if (0 == n % f) {
n /= f;
break;
}
f += 2;
}
if (n != 1) {
}
return n;
}

/**
* Factorization by trial division.
* @param n the number to factor
* @return the list of prime factors of n
*/
public static List<Integer> trialDivision(int n){
final List<Integer> factors = new ArrayList(32);
n = smallTrialDivision(n, factors);
if (1 == n) {
return factors;
}
// here we are sure that n is either a prime or a semi prime
final int bound = (int) FastMath.sqrt(n);
boundedTrialDivision(n, bound, factors);
return factors;
}

/**
* Miller-Rabin probabilistic primality test for int type, used in such a way that a result is always guaranteed.
* <p>
* It uses the prime numbers as successive base therefore it is guaranteed to be always correct.
* (see Handbook of applied cryptography by Menezes, table 4.1)
*
* @param n number to test: an odd integer ≥ 3
* @return true if n is prime. false if n is definitely composite.
*/
public static boolean millerRabinPrimeTest(final int n) {
final int nMinus1 = n - 1;
final int s = Integer.numberOfTrailingZeros(nMinus1);
final int r = nMinus1 >> s;
//r must be odd, it is not checked here
int t = 1;
if (n >= 2047) {
t = 2;
}
if (n >= 1373653) {
t = 3;
}
if (n >= 25326001) {
t = 4;
} // works up to 3.2 billion, int range stops at 2.7 so we are safe :-)
BigInteger br = BigInteger.valueOf(r);
BigInteger bn = BigInteger.valueOf(n);

for (int i = 0; i < t; i++) {
BigInteger a = BigInteger.valueOf(SmallPrimes.PRIMES[i]);
BigInteger bPow = a.modPow(br, bn);
int y = bPow.intValue();
if ((1 != y) && (y != nMinus1)) {
int j = 1;
while ((j <= s - 1) && (nMinus1 != y)) {
long square = ((long) y) * y;
y = (int) (square % n);
if (1 == y) {
return false;
} // definitely composite
j++;
}
if (nMinus1 != y) {
return false;
} // definitely composite
}
}
return true; // definitely prime
}
}

```

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