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A005936
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Pseudoprimes to base 5.
(Formerly M3712)
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18
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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
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1,1
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COMMENTS
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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).
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REFERENCES
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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).
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LINKS
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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