Fermat pseudoprimes

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Revision as of 20:11, 16 August 2022 by Wherrera (talk | contribs) (julia example)
Fermat pseudoprimes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

A Fermat pseudoprime is a positive composite integer that passes the Fermat primality test.

Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p.

For an integer a > 1, if a composite integer x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a.

Fermat pseudoprimes to base 2 are sometimes called Sarrus numbers or Poulet numbers.

Fermat pseudoprimes can be found to any positive integer base. When using a base integer a = 1, this method returns all composite numbers.


Task

For base integers a of 1 through 20:

  • Find the count of pseudoprimes up to and including 12,000.
  • Show the first 20 pseudoprimes.


Stretch
  • Extend the count threshold out to 25,000 or 50,000.


See also


Raku

<lang perl6>use List::Divvy; for 1..20 -> $base { 

   my @pseudo = lazy (1..*).hyper.grep: { !.is-prime && (exp($_ - 1, $base) % $_ == 1) }
   my $threshold = 50000;
   say $base.fmt("Base %2d - Up to $threshold: ") ~ (+@pseudo.&upto: $threshold).fmt('%5d')
       ~ "  First 20: " ~ @pseudo[^20].gist

}</lang> 

Base  1 - Up to 50000: 44866  First 20: (4 6 8 9 10 12 14 15 16 18 20 21 22 24 25 26 27 28 30 32)
Base  2 - Up to 50000:    55  First 20: (341 561 645 1105 1387 1729 1905 2047 2465 2701 2821 3277 4033 4369 4371 4681 5461 6601 7957 8321)
Base  3 - Up to 50000:    53  First 20: (91 121 286 671 703 949 1105 1541 1729 1891 2465 2665 2701 2821 3281 3367 3751 4961 5551 6601)
Base  4 - Up to 50000:   111  First 20: (15 85 91 341 435 451 561 645 703 1105 1247 1271 1387 1581 1695 1729 1891 1905 2047 2071)
Base  5 - Up to 50000:    54  First 20: (4 124 217 561 781 1541 1729 1891 2821 4123 5461 5611 5662 5731 6601 7449 7813 8029 8911 9881)
Base  6 - Up to 50000:    74  First 20: (35 185 217 301 481 1105 1111 1261 1333 1729 2465 2701 2821 3421 3565 3589 3913 4123 4495 5713)
Base  7 - Up to 50000:    49  First 20: (6 25 325 561 703 817 1105 1825 2101 2353 2465 3277 4525 4825 6697 8321 10225 10585 10621 11041)
Base  8 - Up to 50000:   150  First 20: (9 21 45 63 65 105 117 133 153 231 273 341 481 511 561 585 645 651 861 949)
Base  9 - Up to 50000:   113  First 20: (4 8 28 52 91 121 205 286 364 511 532 616 671 697 703 946 949 1036 1105 1288)
Base 10 - Up to 50000:    65  First 20: (9 33 91 99 259 451 481 561 657 703 909 1233 1729 2409 2821 2981 3333 3367 4141 4187)
Base 11 - Up to 50000:    61  First 20: (10 15 70 133 190 259 305 481 645 703 793 1105 1330 1729 2047 2257 2465 2821 4577 4921)
Base 12 - Up to 50000:    91  First 20: (65 91 133 143 145 247 377 385 703 1045 1099 1105 1649 1729 1885 1891 2041 2233 2465 2701)
Base 13 - Up to 50000:    68  First 20: (4 6 12 21 85 105 231 244 276 357 427 561 1099 1785 1891 2465 2806 3605 5028 5149)
Base 14 - Up to 50000:    69  First 20: (15 39 65 195 481 561 781 793 841 985 1105 1111 1541 1891 2257 2465 2561 2665 2743 3277)
Base 15 - Up to 50000:    42  First 20: (14 341 742 946 1477 1541 1687 1729 1891 1921 2821 3133 3277 4187 6541 6601 7471 8701 8911 9073)
Base 16 - Up to 50000:   145  First 20: (15 51 85 91 255 341 435 451 561 595 645 703 1105 1247 1261 1271 1285 1387 1581 1687)
Base 17 - Up to 50000:    63  First 20: (4 8 9 16 45 91 145 261 781 1111 1228 1305 1729 1885 2149 2821 3991 4005 4033 4187)
Base 18 - Up to 50000:    98  First 20: (25 49 65 85 133 221 323 325 343 425 451 637 931 1105 1225 1369 1387 1649 1729 1921)
Base 19 - Up to 50000:    93  First 20: (6 9 15 18 45 49 153 169 343 561 637 889 905 906 1035 1105 1629 1661 1849 1891)
Base 20 - Up to 50000:    66  First 20: (21 57 133 231 399 561 671 861 889 1281 1653 1729 1891 2059 2413 2501 2761 2821 2947 3059)

Julia

<lang ruby>using Primes

ispseudo(n, base) = !isprime(n) && BigInt(base)^(n - 1) % n == 1

for b in 1:20 

   pseudos = filter(n -> ispseudo(n, b), 1:50000)
   println("Base ", lpad(b, 2), " up to 50000: ", lpad(length(pseudos), 5), "  First 20: ", pseudos[1:20])

end

</lang>

Output:
Base  1 up to 50000: 44866  First 20: [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32]
Base  2 up to 50000:    55  First 20: [341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321]      
Base  3 up to 50000:    53  First 20: [91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601]
Base  4 up to 50000:   111  First 20: [15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071]
Base  5 up to 50000:    54  First 20: [4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881]
Base  6 up to 50000:    74  First 20: [35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713]
Base  7 up to 50000:    49  First 20: [6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041]
Base  8 up to 50000:   150  First 20: [9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949]
Base  9 up to 50000:   113  First 20: [4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288]
Base 10 up to 50000:    65  First 20: [9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187]
Base 11 up to 50000:    61  First 20: [10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729, 2047, 2257, 2465, 2821, 4577, 4921]
Base 12 up to 50000:    91  First 20: [65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891, 2041, 2233, 2465, 2701]
Base 13 up to 50000:    68  First 20: [4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891, 2465, 2806, 3605, 5028, 5149]
Base 14 up to 50000:    69  First 20: [15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891, 2257, 2465, 2561, 2665, 2743, 3277]
Base 15 up to 50000:    42  First 20: [14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073]
Base 16 up to 50000:   145  First 20: [15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687]
Base 17 up to 50000:    63  First 20: [4, 8, 9, 16, 45, 91, 145, 261, 781, 1111, 1228, 1305, 1729, 1885, 2149, 2821, 3991, 4005, 4033, 4187]
Base 18 up to 50000:    98  First 20: [25, 49, 65, 85, 133, 221, 323, 325, 343, 425, 451, 637, 931, 1105, 1225, 1369, 1387, 1649, 1729, 1921]
Base 19 up to 50000:    93  First 20: [6, 9, 15, 18, 45, 49, 153, 169, 343, 561, 637, 889, 905, 906, 1035, 1105, 1629, 1661, 1849, 1891]
Base 20 up to 50000:    66 First 20: [21, 57, 133, 231, 399, 561, 671, 861, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2501, 2761, 2821, 2947, 3059]
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