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A266829
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Primes p such that a prime q < p exists with p^(q-1) == 1 (mod q^2) and q^(p-1) == 1 (mod p^2), i.e., primes that are the larger member of a double Wieferich prime pair.
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5
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OFFSET
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1,1
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COMMENTS
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There are no further terms less than 10^6 (cf. Ernvall, Metsänkylä, 1997, p. 1360).
There are no further terms p less than 2^(1/3)*10^10 with p*q <= 10^15 and p and q both odd. (cf. Logan, Mossinghoff, results 4.2.). - Felix Fröhlich, May 29 2016 [Corrected. Felix Fröhlich, Aug 05 2018]
Primes that occur in column 2 of A282293. - Felix Fröhlich, Aug 05 2018
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LINKS
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Table of n, a(n) for n=1..6.
R. Ernvall and T. Metsänkylä, On the p-divisibility of Fermat quotients, Math. Comp., Volume 66, Number 219 (1997), 1353-1365.
B. Logan and M. J. Mossinghoff, Double Wieferich pairs and circulant Hadamard matrices, ResearchGate, 2015.
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MATHEMATICA
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fQ[p_] := Block[{q = 2}, While[q < p && (PowerMod[p, q - 1, q^2] != 1 || PowerMod[q, p - 1, p^2] != 1), q = NextPrime@ q]; If[q < p, True, False]]; p = 3; lst = {}; While[p < 1000000, If[fQ@ p, AppendTo[lst, p]]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Jan 04 2016 *)
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PROG
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(PARI) forprime(p=3, , forprime(q=2, p-1, if(Mod(p, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1, print1(p, ", "); break({1}))))
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CROSSREFS
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Cf. A124122, A282293, A317724 (smallest existing q).
Sequence in context: A331021 A270833 A273471 * A203858 A115192 A307220
Adjacent sequences: A266826 A266827 A266828 * A266830 A266831 A266832
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KEYWORD
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nonn,hard,more
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AUTHOR
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Felix Fröhlich, Jan 04 2016
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EXTENSIONS
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a(5)-a(6) from Felix Fröhlich, May 29 2016
Removed three comments. - Felix Fröhlich, Aug 21 2018
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STATUS
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approved
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