A005936 - OEIS

A005936

Pseudoprimes to base 5.
(Formerly M3712)

4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881, 11041, 11476, 12801, 13021, 13333, 13981, 14981, 15751, 15841, 16297, 17767, 21361, 22791, 23653, 24211, 25327, 25351, 29341, 29539

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OFFSET

1,1

COMMENTS

According to Karsten Meyer, 4 should be excluded, following the strict definition in Crandall and Pomerance. - May 16 2006

Theorem: If both numbers q and (2q - 1) are primes (q is in the sequence A005382) then n = q*(2q - 1) is a pseudoprime to base 5 (n is in the sequence) if and only if q is of the form 10k + 1. 1891, 88831, 146611, 218791, 721801, ... are such terms. This sequence is a subsequence of A122782. - Farideh Firoozbakht, Sep 14 2006

Composite numbers n such that 5^(n-1) == 1 (mod n).

REFERENCES

R. Crandall and C. Pomerance, "Prime Numbers - A Computational Perspective", Second Edition, Springer Verlag 2005, ISBN 0-387-25282-7 Page 132 (Theorem 3.4.2. and Algorithm 3.4.3)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 124, p. 43, Ellipses, Paris 2008.

R. K. Guy, Unsolved Problems in Number Theory, A12.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..92893 (terms a(1)-a(776) from R. J. Mathar, a(777)-a(1000) from T. D. Noe)

J. Bernheiden, Pseudoprimes (Text in German)

C. Pomerance & N. J. A. Sloane, Correspondence, 1991

F. Richman, Primality testing with Fermat's little theorem

Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Index entries for sequences related to pseudoprimes

MATHEMATICA

base = 5; t = {}; n = 1; While[Length[t] < 100, n++; If[! PrimeQ[n] && PowerMod[base, n-1, n] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Feb 21 2012 *)

Select[Range[30000], CompositeQ[#]&&PowerMod[5, #-1, #]==1&] (* Harvey P. Dale, Jul 21 2023 *)

CROSSREFS

Pseudoprimes to other bases: A001567 (2), A005935 (3), A005937 (6), A005938 (7), A005939 (10).

Cf. A005382, A122782.

Sequence in context: A219871 A263550 A232592 * A241648 A197779 A197610