/usr/share/axiom-20170501/src/algebra/PRIMES.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 | )abbrev package PRIMES IntegerPrimesPackage
++ Author: Michael Monagan, James Davenport
++ Date Created: August 1987
++ Date Last Updated: 31 May 1993
++ References:
++ Dave92 Primality Testing Revisited
++ Arna95 Constructing Carmichael numbers which are strong pseudoprimes to
++ several bases
++ Jaes93 On Strong Pseudoprimes to Several Bases
++ Description:
++ The \spadtype{IntegerPrimesPackage} implements a modification of
++ Rabin's probabilistic
++ primality test and the utility functions \spadfun{nextPrime},
++ \spadfun{prevPrime} and \spadfun{primes}.
IntegerPrimesPackage(I) : SIG == CODE where
I : IntegerNumberSystem
SIG ==> with
prime? : I -> Boolean
++ \spad{prime?(n)} returns true if n is prime and false if not.
++ The algorithm used is Rabin's probabilistic primality test
++ (reference: Knuth Volume 2 Semi Numerical Algorithms).
++ If \spad{prime? n} returns false, n is proven composite.
++ If \spad{prime? n} returns true, prime? may be in error
++ however, the probability of error is very low.
++ and is zero below 25*10**9 (due to a result of Pomerance et al),
++ below 10**12 and 10**13 due to results of Pinch,
++ and below 341550071728321 due to a result of Jaeschke.
++ Specifically, this implementation does at least 10 pseudo prime
++ tests and so the probability of error is \spad{< 4**(-10)}.
++ The running time of this method is cubic in the length
++ of the input n, that is \spad{O( (log n)**3 )}, for n<10**20.
++ beyond that, the algorithm is quartic, \spad{O( (log n)**4 )}.
++ Two improvements due to Davenport have been incorporated
++ which catches some trivial strong pseudo-primes, such as
++ [Jaeschke, 1991] 1377161253229053 * 413148375987157, which
++ the original algorithm regards as prime
nextPrime : I -> I
++ \spad{nextPrime(n)} returns the smallest prime strictly larger than n
prevPrime : I -> I
++ \spad{prevPrime(n)} returns the largest prime strictly smaller than n
primes : (I,I) -> List I
++ \spad{primes(a,b)} returns a list of all primes p with
++ \spad{a <= p <= b}
CODE ==> add
smallPrimes: List I :=
[2::I, 3::I, 5::I, 7::I, 11::I, 13::I, 17::I, 19::I,_
23::I, 29::I, 31::I, 37::I, 41::I, 43::I, 47::I, 53::I,_
59::I, 61::I, 67::I, 71::I, 73::I, 79::I, 83::I, 89::I,_
97::I, 101::I, 103::I, 107::I, 109::I, 113::I, 127::I,_
131::I, 137::I, 139::I, 149::I, 151::I, 157::I, 163::I,_
167::I, 173::I, 179::I, 181::I, 191::I, 193::I, 197::I,_
199::I, 211::I, 223::I, 227::I, 229::I, 233::I, 239::I,_
241::I, 251::I, 257::I, 263::I, 269::I, 271::I, 277::I,_
281::I, 283::I, 293::I, 307::I, 311::I, 313::I, 317::I,_
331::I, 337::I, 347::I, 349::I, 353::I, 359::I, 367::I,_
373::I, 379::I, 383::I, 389::I, 397::I, 401::I, 409::I,_
419::I, 421::I, 431::I, 433::I, 439::I, 443::I, 449::I,_
457::I, 461::I, 463::I, 467::I, 479::I, 487::I, 491::I,_
499::I, 503::I, 509::I, 521::I, 523::I, 541::I, 547::I,_
557::I, 563::I, 569::I, 571::I, 577::I, 587::I, 593::I,_
599::I, 601::I, 607::I, 613::I, 617::I, 619::I, 631::I,_
641::I, 643::I, 647::I, 653::I, 659::I, 661::I, 673::I,_
677::I, 683::I, 691::I, 701::I, 709::I, 719::I, 727::I,_
733::I, 739::I, 743::I, 751::I, 757::I, 761::I, 769::I,_
773::I, 787::I, 797::I, 809::I, 811::I, 821::I, 823::I,_
827::I, 829::I, 839::I, 853::I, 857::I, 859::I, 863::I,_
877::I, 881::I, 883::I, 887::I, 907::I, 911::I, 919::I,_
929::I, 937::I, 941::I, 947::I, 953::I, 967::I, 971::I,_
977::I, 983::I, 991::I, 997::I, 1009::I, 1013::I,_
1019::I, 1021::I, 1031::I, 1033::I, 1039::I, 1049::I,_
1051::I, 1061::I, 1063::I, 1069::I, 1087::I, 1091::I,_
1093::I, 1097::I, 1103::I, 1109::I, 1117::I, 1123::I,_
1129::I, 1151::I, 1153::I, 1163::I, 1171::I, 1181::I,_
1187::I, 1193::I, 1201::I, 1213::I, 1217::I, 1223::I,_
1229::I, 1231::I, 1237::I, 1249::I, 1259::I, 1277::I,_
1279::I, 1283::I, 1289::I, 1291::I, 1297::I, 1301::I,_
1303::I, 1307::I, 1319::I, 1321::I, 1327::I, 1361::I,_
1367::I, 1373::I, 1381::I, 1399::I, 1409::I, 1423::I,_
1427::I, 1429::I, 1433::I, 1439::I, 1447::I, 1451::I,_
1453::I, 1459::I, 1471::I, 1481::I, 1483::I, 1487::I,_
1489::I, 1493::I, 1499::I, 1511::I, 1523::I, 1531::I,_
1543::I, 1549::I, 1553::I, 1559::I, 1567::I, 1571::I,_
1579::I, 1583::I, 1597::I, 1601::I, 1607::I, 1609::I,_
1613::I, 1619::I, 1621::I, 1627::I, 1637::I, 1657::I,_
1663::I, 1667::I, 1669::I, 1693::I, 1697::I, 1699::I,_
1709::I, 1721::I, 1723::I, 1733::I, 1741::I, 1747::I,_
1753::I, 1759::I, 1777::I, 1783::I, 1787::I, 1789::I,_
1801::I, 1811::I, 1823::I, 1831::I, 1847::I, 1861::I,_
1867::I, 1871::I, 1873::I, 1877::I, 1879::I, 1889::I,_
1901::I, 1907::I, 1913::I, 1931::I, 1933::I, 1949::I,_
1951::I, 1973::I, 1979::I, 1987::I, 1993::I, 1997::I,_
1999::I, 2003::I, 2011::I, 2017::I, 2027::I, 2029::I,_
2039::I, 2053::I, 2063::I, 2069::I, 2081::I, 2083::I,_
2087::I, 2089::I, 2099::I, 2111::I, 2113::I, 2129::I,_
2131::I, 2137::I, 2141::I, 2143::I, 2153::I, 2161::I,_
2179::I, 2203::I, 2207::I, 2213::I, 2221::I, 2237::I,_
2239::I, 2243::I, 2251::I, 2267::I, 2269::I, 2273::I,_
2281::I, 2287::I, 2293::I, 2297::I, 2309::I, 2311::I,_
2333::I, 2339::I, 2341::I, 2347::I, 2351::I, 2357::I,_
2371::I, 2377::I, 2381::I, 2383::I, 2389::I, 2393::I,_
2399::I, 2411::I, 2417::I, 2423::I, 2437::I, 2441::I,_
2447::I, 2459::I, 2467::I, 2473::I, 2477::I, 2503::I,_
2521::I, 2531::I, 2539::I, 2543::I, 2549::I, 2551::I,_
2557::I, 2579::I, 2591::I, 2593::I, 2609::I, 2617::I,_
2621::I, 2633::I, 2647::I, 2657::I, 2659::I, 2663::I,_
2671::I, 2677::I, 2683::I, 2687::I, 2689::I, 2693::I,_
2699::I, 2707::I, 2711::I, 2713::I, 2719::I, 2729::I,_
2731::I, 2741::I, 2749::I, 2753::I, 2767::I, 2777::I,_
2789::I, 2791::I, 2797::I, 2801::I, 2803::I, 2819::I,_
2833::I, 2837::I, 2843::I, 2851::I, 2857::I, 2861::I,_
2879::I, 2887::I, 2897::I, 2903::I, 2909::I, 2917::I,_
2927::I, 2939::I, 2953::I, 2957::I, 2963::I, 2969::I,_
2971::I, 2999::I, 3001::I, 3011::I, 3019::I, 3023::I,_
3037::I, 3041::I, 3049::I, 3061::I, 3067::I, 3079::I,_
3083::I, 3089::I, 3109::I, 3119::I, 3121::I, 3137::I,_
3163::I, 3167::I, 3169::I, 3181::I, 3187::I, 3191::I,_
3203::I, 3209::I, 3217::I, 3221::I, 3229::I, 3251::I,_
3253::I, 3257::I, 3259::I, 3271::I, 3299::I, 3301::I,_
3307::I, 3313::I, 3319::I, 3323::I, 3329::I, 3331::I,_
3343::I, 3347::I, 3359::I, 3361::I, 3371::I, 3373::I,_
3389::I, 3391::I, 3407::I, 3413::I, 3433::I, 3449::I,_
3457::I, 3461::I, 3463::I, 3467::I, 3469::I, 3491::I,_
3499::I, 3511::I, 3517::I, 3527::I, 3529::I, 3533::I,_
3539::I, 3541::I, 3547::I, 3557::I, 3559::I, 3571::I,_
3581::I, 3583::I, 3593::I, 3607::I, 3613::I, 3617::I,_
3623::I, 3631::I, 3637::I, 3643::I, 3659::I, 3671::I,_
3673::I, 3677::I, 3691::I, 3697::I, 3701::I, 3709::I,_
3719::I, 3727::I, 3733::I, 3739::I, 3761::I, 3767::I,_
3769::I, 3779::I, 3793::I, 3797::I, 3803::I, 3821::I,_
3823::I, 3833::I, 3847::I, 3851::I, 3853::I, 3863::I,_
3877::I, 3881::I, 3889::I, 3907::I, 3911::I, 3917::I,_
3919::I, 3923::I, 3929::I, 3931::I, 3943::I, 3947::I,_
3967::I, 3989::I, 4001::I, 4003::I, 4007::I, 4013::I,_
4019::I, 4021::I, 4027::I, 4049::I, 4051::I, 4057::I,_
4073::I, 4079::I, 4091::I, 4093::I, 4099::I, 4111::I,_
4127::I, 4129::I, 4133::I, 4139::I, 4153::I, 4157::I,_
4159::I, 4177::I, 4201::I, 4211::I, 4217::I, 4219::I,_
4229::I, 4231::I, 4241::I, 4243::I, 4253::I, 4259::I,_
4261::I, 4271::I, 4273::I, 4283::I, 4289::I, 4297::I,_
4327::I, 4337::I, 4339::I, 4349::I, 4357::I, 4363::I,_
4373::I, 4391::I, 4397::I, 4409::I, 4421::I, 4423::I,_
4441::I, 4447::I, 4451::I, 4457::I, 4463::I, 4481::I,_
4483::I, 4493::I, 4507::I, 4513::I, 4517::I, 4519::I,_
4523::I, 4547::I, 4549::I, 4561::I, 4567::I, 4583::I,_
4591::I, 4597::I, 4603::I, 4621::I, 4637::I, 4639::I,_
4643::I, 4649::I, 4651::I, 4657::I, 4663::I, 4673::I,_
4679::I, 4691::I, 4703::I, 4721::I, 4723::I, 4729::I,_
4733::I, 4751::I, 4759::I, 4783::I, 4787::I, 4789::I,_
4793::I, 4799::I, 4801::I, 4813::I, 4817::I, 4831::I,_
4861::I, 4871::I, 4877::I, 4889::I, 4903::I, 4909::I,_
4919::I, 4931::I, 4933::I, 4937::I, 4943::I, 4951::I,_
4957::I, 4967::I, 4969::I, 4973::I, 4987::I, 4993::I,_
4999::I, 5003::I, 5009::I, 5011::I, 5021::I, 5023::I,_
5039::I, 5051::I, 5059::I, 5077::I, 5081::I, 5087::I,_
5099::I, 5101::I, 5107::I, 5113::I, 5119::I, 5147::I,_
5153::I, 5167::I, 5171::I, 5179::I, 5189::I, 5197::I,_
5209::I, 5227::I, 5231::I, 5233::I, 5237::I, 5261::I,_
5273::I, 5279::I, 5281::I, 5297::I, 5303::I, 5309::I,_
5323::I, 5333::I, 5347::I, 5351::I, 5381::I, 5387::I,_
5393::I, 5399::I, 5407::I, 5413::I, 5417::I, 5419::I,_
5431::I, 5437::I, 5441::I, 5443::I, 5449::I, 5471::I,_
5477::I, 5479::I, 5483::I, 5501::I, 5503::I, 5507::I,_
5519::I, 5521::I, 5527::I, 5531::I, 5557::I, 5563::I,_
5569::I, 5573::I, 5581::I, 5591::I, 5623::I, 5639::I,_
5641::I, 5647::I, 5651::I, 5653::I, 5657::I, 5659::I,_
5669::I, 5683::I, 5689::I, 5693::I, 5701::I, 5711::I,_
5717::I, 5737::I, 5741::I, 5743::I, 5749::I, 5779::I,_
5783::I, 5791::I, 5801::I, 5807::I, 5813::I, 5821::I,_
5827::I, 5839::I, 5843::I, 5849::I, 5851::I, 5857::I,_
5861::I, 5867::I, 5869::I, 5879::I, 5881::I, 5897::I,_
5903::I, 5923::I, 5927::I, 5939::I, 5953::I, 5981::I,_
5987::I, 6007::I, 6011::I, 6029::I, 6037::I, 6043::I,_
6047::I, 6053::I, 6067::I, 6073::I, 6079::I, 6089::I,_
6091::I, 6101::I, 6113::I, 6121::I, 6131::I, 6133::I,_
6143::I, 6151::I, 6163::I, 6173::I, 6197::I, 6199::I,_
6203::I, 6211::I, 6217::I, 6221::I, 6229::I, 6247::I,_
6257::I, 6263::I, 6269::I, 6271::I, 6277::I, 6287::I,_
6299::I, 6301::I, 6311::I, 6317::I, 6323::I, 6329::I,_
6337::I, 6343::I, 6353::I, 6359::I, 6361::I, 6367::I,_
6373::I, 6379::I, 6389::I, 6397::I, 6421::I, 6427::I,_
6449::I, 6451::I, 6469::I, 6473::I, 6481::I, 6491::I,_
6521::I, 6529::I, 6547::I, 6551::I, 6553::I, 6563::I,_
6569::I, 6571::I, 6577::I, 6581::I, 6599::I, 6607::I,_
6619::I, 6637::I, 6653::I, 6659::I, 6661::I, 6673::I,_
6679::I, 6689::I, 6691::I, 6701::I, 6703::I, 6709::I,_
6719::I, 6733::I, 6737::I, 6761::I, 6763::I, 6779::I,_
6781::I, 6791::I, 6793::I, 6803::I, 6823::I, 6827::I,_
6829::I, 6833::I, 6841::I, 6857::I, 6863::I, 6869::I,_
6871::I, 6883::I, 6899::I, 6907::I, 6911::I, 6917::I,_
6947::I, 6949::I, 6959::I, 6961::I, 6967::I, 6971::I,_
6977::I, 6983::I, 6991::I, 6997::I, 7001::I, 7013::I,_
7019::I, 7027::I, 7039::I, 7043::I, 7057::I, 7069::I,_
7079::I, 7103::I, 7109::I, 7121::I, 7127::I, 7129::I,_
7151::I, 7159::I, 7177::I, 7187::I, 7193::I, 7207::I,_
7211::I, 7213::I, 7219::I, 7229::I, 7237::I, 7243::I,_
7247::I, 7253::I, 7283::I, 7297::I, 7307::I, 7309::I,_
7321::I, 7331::I, 7333::I, 7349::I, 7351::I, 7369::I,_
7393::I, 7411::I, 7417::I, 7433::I, 7451::I, 7457::I,_
7459::I, 7477::I, 7481::I, 7487::I, 7489::I, 7499::I,_
7507::I, 7517::I, 7523::I, 7529::I, 7537::I, 7541::I,_
7547::I, 7549::I, 7559::I, 7561::I, 7573::I, 7577::I,_
7583::I, 7589::I, 7591::I, 7603::I, 7607::I, 7621::I,_
7639::I, 7643::I, 7649::I, 7669::I, 7673::I, 7681::I,_
7687::I, 7691::I, 7699::I, 7703::I, 7717::I, 7723::I,_
7727::I, 7741::I, 7753::I, 7757::I, 7759::I, 7789::I,_
7793::I, 7817::I, 7823::I, 7829::I, 7841::I, 7853::I,_
7867::I, 7873::I, 7877::I, 7879::I, 7883::I, 7901::I,_
7907::I, 7919::I, 7927::I, 7933::I, 7937::I, 7949::I,_
7951::I, 7963::I, 7993::I, 8009::I, 8011::I, 8017::I,_
8039::I, 8053::I, 8059::I, 8069::I, 8081::I, 8087::I,_
8089::I, 8093::I, 8101::I, 8111::I, 8117::I, 8123::I,_
8147::I, 8161::I, 8167::I, 8171::I, 8179::I, 8191::I,_
8209::I, 8219::I, 8221::I, 8231::I, 8233::I, 8237::I,_
8243::I, 8263::I, 8269::I, 8273::I, 8287::I, 8291::I,_
8293::I, 8297::I, 8311::I, 8317::I, 8329::I, 8353::I,_
8363::I, 8369::I, 8377::I, 8387::I, 8389::I, 8419::I,_
8423::I, 8429::I, 8431::I, 8443::I, 8447::I, 8461::I,_
8467::I, 8501::I, 8513::I, 8521::I, 8527::I, 8537::I,_
8539::I, 8543::I, 8563::I, 8573::I, 8581::I, 8597::I,_
8599::I, 8609::I, 8623::I, 8627::I, 8629::I, 8641::I,_
8647::I, 8663::I, 8669::I, 8677::I, 8681::I, 8689::I,_
8693::I, 8699::I, 8707::I, 8713::I, 8719::I, 8731::I,_
8737::I, 8741::I, 8747::I, 8753::I, 8761::I, 8779::I,_
8783::I, 8803::I, 8807::I, 8819::I, 8821::I, 8831::I,_
8837::I, 8839::I, 8849::I, 8861::I, 8863::I, 8867::I,_
8887::I, 8893::I, 8923::I, 8929::I, 8933::I, 8941::I,_
8951::I, 8963::I, 8969::I, 8971::I, 8999::I, 9001::I,_
9007::I, 9011::I, 9013::I, 9029::I, 9041::I, 9043::I,_
9049::I, 9059::I, 9067::I, 9091::I, 9103::I, 9109::I,_
9127::I, 9133::I, 9137::I, 9151::I, 9157::I, 9161::I,_
9173::I, 9181::I, 9187::I, 9199::I, 9203::I, 9209::I,_
9221::I, 9227::I, 9239::I, 9241::I, 9257::I, 9277::I,_
9281::I, 9283::I, 9293::I, 9311::I, 9319::I, 9323::I,_
9337::I, 9341::I, 9343::I, 9349::I, 9371::I, 9377::I,_
9391::I, 9397::I, 9403::I, 9413::I, 9419::I, 9421::I,_
9431::I, 9433::I, 9437::I, 9439::I, 9461::I, 9463::I,_
9467::I, 9473::I, 9479::I, 9491::I, 9497::I, 9511::I,_
9521::I, 9533::I, 9539::I, 9547::I, 9551::I, 9587::I,_
9601::I, 9613::I, 9619::I, 9623::I, 9629::I, 9631::I,_
9643::I, 9649::I, 9661::I, 9677::I, 9679::I, 9689::I,_
9697::I, 9719::I, 9721::I, 9733::I, 9739::I, 9743::I,_
9749::I, 9767::I, 9769::I, 9781::I, 9787::I, 9791::I,_
9803::I, 9811::I, 9817::I, 9829::I, 9833::I, 9839::I,_
9851::I, 9857::I, 9859::I, 9871::I, 9883::I, 9887::I,_
9901::I, 9907::I, 9923::I, 9929::I, 9931::I, 9941::I,_
9949::I, 9967::I, 9973::I]
productSmallPrimes := */smallPrimes
nextSmallPrime := 10007::I
nextSmallPrimeSquared := nextSmallPrime**2
two := 2::I
tenPowerTwenty:=(10::I)**20
PomeranceList:= [25326001::I, 161304001::I, 960946321::I, 1157839381::I,
-- 3215031751::I, -- has a factor of 151
3697278427::I, 5764643587::I, 6770862367::I,
14386156093::I, 15579919981::I, 18459366157::I,
19887974881::I, 21276028621::I ]::(List I)
PomeranceLimit:=27716349961::I -- replaces (25*10**9) due to Pinch
PinchList:= _
[3215031751::I, 118670087467::I, 128282461501::I, 354864744877::I,
546348519181::I, 602248359169::I, 669094855201::I ]
PinchLimit:= (10**12)::I
PinchList2:= [2152302898747::I, 3474749660383::I]
PinchLimit2:= (10**13)::I
JaeschkeLimit:=341550071728321::I
rootsMinus1:Set I := empty()
-- used to check whether we detect too many roots of -1
count2Order:Vector NonNegativeInteger := new(1,0)
-- used to check whether we observe an element of maximal two-order
primes(m, n) ==
-- computes primes from m to n inclusive using prime?
l:List(I) :=
m <= two => [two]
empty()
n < two or n < m => empty()
if even? m then m := m + 1
ll:List(I) := [k::I for k in
convert(m)@Integer..convert(n)@Integer by 2 | prime?(k::I)]
reverse_! concat_!(ll, l)
rabinProvesComposite : (I,I,I,I,NonNegativeInteger) -> Boolean
rabinProvesCompositeSmall : (I,I,I,I,NonNegativeInteger) -> Boolean
rabinProvesCompositeSmall(p,n,nm1,q,k) ==
-- probability n prime is > 3/4 for each iteration
-- for most n this probability is much greater than 3/4
t := powmod(p, q, n)
-- neither of these cases tells us anything
if not ((t = 1) or t = nm1) then
for j in 1..k-1 repeat
oldt := t
t := mulmod(t, t, n)
(t = 1) => return true
-- we have squared someting not -1 and got 1
t = nm1 =>
leave
not (t = nm1) => return true
false
rabinProvesComposite(p,n,nm1,q,k) ==
-- probability n prime is > 3/4 for each iteration
-- for most n this probability is much greater than 3/4
t := powmod(p, q, n)
-- neither of these cases tells us anything
if t=nm1 then count2Order(1):=count2Order(1)+1
if not ((t = 1) or t = nm1) then
for j in 1..k-1 repeat
oldt := t
t := mulmod(t, t, n)
(t = 1) => return true
-- we have squared someting not -1 and got 1
t = nm1 =>
rootsMinus1:=union(rootsMinus1,oldt)
count2Order(j+1):=count2Order(j+1)+1
leave
not (t = nm1) => return true
# rootsMinus1 > 2 => true -- Z/nZ can't be a field
false
prime? n ==
n < two => false
n < nextSmallPrime => member?(n, smallPrimes)
not (gcd(n, productSmallPrimes) = 1) => false
n < nextSmallPrimeSquared => true
nm1 := n-1
q := (nm1) quo two
for k in 1.. while not odd? q repeat q := q quo two
-- q = (n-1) quo 2**k for largest possible k
n < JaeschkeLimit =>
rabinProvesCompositeSmall(2::I,n,nm1,q,k) => return false
rabinProvesCompositeSmall(3::I,n,nm1,q,k) => return false
n < PomeranceLimit =>
rabinProvesCompositeSmall(5::I,n,nm1,q,k) => return false
member?(n,PomeranceList) => return false
true
rabinProvesCompositeSmall(7::I,n,nm1,q,k) => return false
n < PinchLimit =>
rabinProvesCompositeSmall(10::I,n,nm1,q,k) => return false
member?(n,PinchList) => return false
true
rabinProvesCompositeSmall(5::I,n,nm1,q,k) => return false
rabinProvesCompositeSmall(11::I,n,nm1,q,k) => return false
n < PinchLimit2 =>
member?(n,PinchList2) => return false
true
rabinProvesCompositeSmall(13::I,n,nm1,q,k) => return false
rabinProvesCompositeSmall(17::I,n,nm1,q,k) => return false
true
rootsMinus1:= empty()
count2Order := new(k,0) -- vector of k zeroes
mn := minIndex smallPrimes
for i in mn+1..mn+10 repeat
rabinProvesComposite(smallPrimes i,n,nm1,q,k) => return false
import IntegerRoots(I)
q > 1 and perfectSquare?(3*n+1) => false
((n9:=n rem (9::I))=1 or n9 = -1) and perfectSquare?(8*n+1) => false
-- Both previous tests from Damgard & Landrock
currPrime:=smallPrimes(mn+10)
probablySafe:=tenPowerTwenty
while count2Order(k) = 0 or n > probablySafe repeat
currPrime := nextPrime currPrime
probablySafe:=probablySafe*(100::I)
rabinProvesComposite(currPrime,n,nm1,q,k) => return false
true
nextPrime n ==
-- computes the first prime after n
n < two => two
if odd? n then n := n + two else n := n + 1
while not prime? n repeat n := n + two
n
prevPrime n ==
-- computes the first prime before n
n < 3::I => error "no primes less than 2"
n = 3::I => two
if odd? n then n := n - two else n := n - 1
while not prime? n repeat n := n - two
n
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