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| 1 | package prime; | |
| 2 | ||
| 3 | import java.io.PrintStream; | |
| 4 | import java.util.ArrayList; | |
| 5 | import java.util.Arrays; | |
| 6 | import java.util.Collection; | |
| 7 | ||
| 8 | /** | |
| 9 | * A prime number utility for generating X number of primes, determining whether | |
| 10 | * a given number is prime, and computing the prime factorization for a number. | |
| 11 | * | |
| 12 | * For determining whether a number N is prime, you only need to know all the | |
| 13 | * primes up to and including sqrt(N). | |
| 14 | * | |
| 15 | * @author Philip Diffenderfer | |
| 16 | * | |
| 17 | */ | |
| 18 | public class Prime | |
| 19 | {
| |
| 20 | ||
| 21 | public static void main( String[] args ) | |
| 22 | {
| |
| 23 | - | Prime p = new Prime( 10000000 ); |
| 23 | + | Prime p = new Prime( 100000 ); |
| 24 | ||
| 25 | // Load 100,000 primes into cache | |
| 26 | long t0 = System.nanoTime(); | |
| 27 | - | p.initialize( 10000000 ); |
| 27 | + | p.initialize( 100000 ); |
| 28 | ||
| 29 | System.out.println( ( System.nanoTime() - t0 ) * 0.000000001 + " seconds to initialize " + (p.length + 1) + " primes" ); | |
| 30 | ||
| 31 | // Last Prime | |
| 32 | System.out.println( p.primes[p.length] ); | |
| 33 | ||
| 34 | // Prime Factors | |
| 35 | System.out.println( p.factorize( 83475L, new ArrayList<Long>() ) ); | |
| 36 | ||
| 37 | // 100 primes | |
| 38 | p.print( System.out, 100 ); | |
| 39 | } | |
| 40 | ||
| 41 | // The cache of prime numbers | |
| 42 | private long[] primes; | |
| 43 | ||
| 44 | // The index of the last prime number in the cache | |
| 45 | private int length; | |
| 46 | ||
| 47 | /** | |
| 48 | * Instantiates a new Prime class with a maximum number of primes it can | |
| 49 | * generate. | |
| 50 | * | |
| 51 | * @param primeCountMax | |
| 52 | * The maximum number of primes this class may be able to store. | |
| 53 | */ | |
| 54 | public Prime( int primeCountMax ) | |
| 55 | {
| |
| 56 | primes = new long[primeCountMax]; | |
| 57 | primes[0] = 2; | |
| 58 | primes[1] = 3; | |
| 59 | primes[2] = 5; | |
| 60 | length = 2; | |
| 61 | } | |
| 62 | ||
| 63 | /** | |
| 64 | * Prints out all primes in the prime cache one line at a time into the | |
| 65 | * PrintStream. | |
| 66 | * | |
| 67 | - | public void print( PrintStream out ) |
| 67 | + | |
| 68 | * The PrintStream to print the prime cache out to. | |
| 69 | */ | |
| 70 | public void print( PrintStream out, int n ) | |
| 71 | {
| |
| 72 | for (int i = 0; i <= n; i++) | |
| 73 | {
| |
| 74 | out.println( primes[i] ); | |
| 75 | } | |
| 76 | } | |
| 77 | ||
| 78 | /** | |
| 79 | * Determines whether the given number is prime. | |
| 80 | * | |
| 81 | * @param n | |
| 82 | * The number to check for primeness. | |
| 83 | * @return True if the number is prime, otherwise false. | |
| 84 | */ | |
| 85 | public boolean isPrime( long n ) | |
| 86 | {
| |
| 87 | // Does n exist in the prime cache? | |
| 88 | if ( isQuickPrimeApplicable( n ) ) | |
| 89 | {
| |
| 90 | // Perform a quick search for n. | |
| 91 | return isQuickPrime( n ); | |
| 92 | } | |
| 93 | ||
| 94 | long max = (long) Math.sqrt( n ); | |
| 95 | ||
| 96 | // Ensure we have enough primes in the prime cache to determine whether | |
| 97 | // or not n is a prime. | |
| 98 | fillPrimes( max ); | |
| 99 | ||
| 100 | // If itself is it's only factor, it's a prime. | |
| 101 | return getFactor( n, max ) == n; | |
| 102 | } | |
| 103 | ||
| 104 | /** | |
| 105 | * Returns the first prime factor for n, or n if n is prime. | |
| 106 | * | |
| 107 | * @param n | |
| 108 | * The number to find the first prime factor of. | |
| 109 | * @param nsqrt | |
| 110 | * The square-root of n. | |
| 111 | * @return The first prime factor of n, or n if it's prime. | |
| 112 | */ | |
| 113 | private long getFactor( long n, long nsqrt ) | |
| 114 | {
| |
| 115 | for (int i = 0; i <= length; i++) | |
| 116 | {
| |
| 117 | long p = primes[i]; | |
| 118 | ||
| 119 | if ( p > nsqrt ) | |
| 120 | {
| |
| 121 | return n; | |
| 122 | } | |
| 123 | ||
| 124 | if ( n % p == 0 ) | |
| 125 | {
| |
| 126 | return p; | |
| 127 | } | |
| 128 | } | |
| 129 | ||
| 130 | return n; | |
| 131 | } | |
| 132 | ||
| 133 | /** | |
| 134 | * Ensures the prime cache has all primes up to some number max. | |
| 135 | * | |
| 136 | * @param max | |
| 137 | * The number the prime cache should encompass (have primes <=) | |
| 138 | */ | |
| 139 | public void fillPrimes( long max ) | |
| 140 | {
| |
| 141 | while ( primes[length] < max ) | |
| 142 | {
| |
| 143 | long p = nextPrime( length ); | |
| 144 | ||
| 145 | primes[++length] = p; | |
| 146 | } | |
| 147 | } | |
| 148 | ||
| 149 | /** | |
| 150 | * Ensures the prime cache has at least primeCount primes. | |
| 151 | * | |
| 152 | * @param primeCount | |
| 153 | * The minimum number of primes the prime cache should have. | |
| 154 | */ | |
| 155 | public void initialize( int primeCount ) | |
| 156 | {
| |
| 157 | --primeCount; | |
| 158 | ||
| 159 | while ( length < primeCount ) | |
| 160 | {
| |
| 161 | long p = nextPrime( length ); | |
| 162 | ||
| 163 | primes[++length] = p; | |
| 164 | } | |
| 165 | } | |
| 166 | ||
| 167 | /** | |
| 168 | * Computes the next prime in the sequence of all primes based on the prime | |
| 169 | * at the given index in the prime cache. | |
| 170 | * | |
| 171 | * @param n | |
| 172 | * The index of a prime in the prime cache, the prime this method | |
| 173 | * returns will be the next prime in the sequence. | |
| 174 | * @return The next prime in the sequence of primes. | |
| 175 | - | long x = primes[n] + jump( n ); |
| 175 | + | |
| 176 | public long nextPrime( int n ) | |
| 177 | {
| |
| 178 | long x = primes[n] + 2; | |
| 179 | - | x += jump( ++n ); |
| 179 | + | |
| 180 | while ( getFactor( x, (long) Math.sqrt( x ) ) != x ) | |
| 181 | {
| |
| 182 | x += 2; | |
| 183 | } | |
| 184 | ||
| 185 | return x; | |
| 186 | } | |
| 187 | ||
| 188 | /** | |
| 189 | * A number is a quick prime if it exists in the currently generated cache | |
| 190 | * of primes. This can efficiently be determined with a binary search. | |
| 191 | * | |
| 192 | * @param n | |
| 193 | * The prime to check. | |
| 194 | * @return True if the given number exists in the prime cache. | |
| 195 | */ | |
| 196 | public boolean isQuickPrime( long n ) | |
| 197 | {
| |
| 198 | return Arrays.binarySearch( primes, 0, length + 1, n ) >= 0; | |
| 199 | } | |
| 200 | ||
| 201 | /** | |
| 202 | * A number is quick prime applicable if there exists a prime in the cache | |
| 203 | * that is larger than the provided number. This implies a search can be | |
| 204 | * done in the array to determine if the provided number is a prime. | |
| 205 | * | |
| 206 | * @param n | |
| 207 | * The number to check for quick prime applicability. | |
| 208 | * @return True if the number exists on the prime cache, otherwise false. | |
| 209 | */ | |
| 210 | public boolean isQuickPrimeApplicable( long n ) | |
| 211 | {
| |
| 212 | return n <= primes[length]; | |
| 213 | - | * Given the index of a prime, return the jump (either 2 or 4) to use to |
| 213 | + | |
| 214 | - | * approximate the next prime. The first 3 primes are excluded from this |
| 214 | + | |
| 215 | - | * rule. |
| 215 | + | |
| 216 | * Adds all prime factors of n to the given collection of numbers. | |
| 217 | - | * @param x |
| 217 | + | |
| 218 | - | * The index of the prime. |
| 218 | + | |
| 219 | - | * @return The amount to jump to the next possible prime. |
| 219 | + | |
| 220 | * @param primeFactors | |
| 221 | - | private int jump( int x ) |
| 221 | + | |
| 222 | * @return The collection given. | |
| 223 | - | return ( 1 << ( ( x & 1 ) + 1 ) ); |
| 223 | + | |
| 224 | public <D extends Collection<Long>> D factorize( long n, D primeFactors ) | |
| 225 | {
| |
| 226 | if ( isQuickPrimeApplicable( n ) && isQuickPrime( n ) ) | |
| 227 | {
| |
| 228 | return primeFactors; | |
| 229 | } | |
| 230 | ||
| 231 | long max = (long) Math.sqrt( n ); | |
| 232 | ||
| 233 | fillPrimes( max ); | |
| 234 | ||
| 235 | while ( n != 1 ) | |
| 236 | {
| |
| 237 | long f = getFactor( n, max ); | |
| 238 | ||
| 239 | primeFactors.add( f ); | |
| 240 | ||
| 241 | n /= f; | |
| 242 | ||
| 243 | max = (long) Math.sqrt( n ); | |
| 244 | } | |
| 245 | ||
| 246 | return primeFactors; | |
| 247 | } | |
| 248 | ||
| 249 | } |