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%I M3712 #46 Jul 21 2023 12:52:20
%S 4,124,217,561,781,1541,1729,1891,2821,4123,5461,5611,5662,5731,6601,
%T 7449,7813,8029,8911,9881,11041,11476,12801,13021,13333,13981,14981,
%U 15751,15841,16297,17767,21361,22791,23653,24211,25327,25351,29341,29539
%N Pseudoprimes to base 5.
%C According to Karsten Meyer, 4 should be excluded, following the strict definition in Crandall and Pomerance. - May 16 2006
%C 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
%C Composite numbers n such that 5^(n-1) == 1 (mod n).
%D 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)
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 124, p. 43, Ellipses, Paris 2008.
%D R. K. Guy, Unsolved Problems in Number Theory, A12.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, <a href="/A005936/b005936.txt">Table of n, a(n) for n = 1..92893</a> (terms a(1)-a(776) from R. J. Mathar, a(777)-a(1000) from T. D. Noe)
%H J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/PSP.pdf">Pseudoprimes (Text in German)</a>
%H C. Pomerance & N. J. A. Sloane, <a href="/A001567/a001567_4.pdf">Correspondence, 1991</a>
%H F. Richman, <a href="http://math.fau.edu/Richman/carm.htm">Primality testing with Fermat's little theorem</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPseudoprime.html">Fermat Pseudoprime</a>
%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>
%t 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 *)
%t Select[Range[30000],CompositeQ[#]&&PowerMod[5,#-1,#]==1&] (* _Harvey P. Dale_, Jul 21 2023 *)
%Y Pseudoprimes to other bases: A001567 (2), A005935 (3), A005937 (6), A005938 (7), A005939 (10).
%Y Cf. A005382, A122782.
%K nonn
%O 1,1
%A _N. J. A. Sloane_.
%E More terms from _David W. Wilson_, Aug 15 1996
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