A006970 Euler pseudoprimes: composite numbers n such that 2^((n-1)/2) == +-1 (mod n).

341, 561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 5461, 6601, 8321, 8481, 10261, 10585, 12801, 15709, 15841, 16705, 18705, 25761, 29341, 30121, 31621, 33153, 34945, 41041, 42799

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v, Richard Pinch

Keywords

• nonn
• nice

Comments

Pseudoprimes for the primality test from [Schick]: n odd is probably prime if (n-1) | A003558((n-1)/2). (Succeeds for 99.9975% of odd natural numbers less than 10^8.) - Jonathan Skowera, Jun 29 2013

Equivalently, these are composites n such that ((n-1)/2)^((n-1)/2) == +-1 (mod n). - Thomas Ordowski, Nov 28 2023

References

• R. K. Guy, Unsolved Problems in Number Theory, A12.
• C. Schick, Weiche Primzahlen und das 257-Eck, 2008, pages 140-146.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1231 from T. D. Noe)
• Eric Weisstein's World of Mathematics, Euler Pseudoprime.
• Index entries for sequences related to pseudoprimes

Programs

Mathematica

ok[?PrimeQ] = False; ok[n] := (p = PowerMod[2, (n - 1)/2, n]; p == Mod[1, n] || p == Mod[-1, n]); Select[2 Range[22000] + 1, ok] (* Jean-François Alcover, Apr 06 2011 *)

PARI

isok(n) = {if (!isprime(n) && (n%2), npm = Mod(2, n)^((n-1)/2); if ((npm == Mod(1,n)) || (npm == Mod(-1,n)), print1(n, ", ")););} \\ Michel Marcus, Sep 12 2015

Extensions

a(15) corrected (to 10261 from 10241) by Faron Moller (fm(AT)csd.uu.se)

Name edited by Thomas Ordowski, Nov 28 2023