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A266829 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. 5

%I

%S 1093,4871,18787,318917,1006003,1645333507

%N 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.

%C There are no further terms less than 10^6 (cf. Ernvall, Metsänkylä, 1997, p. 1360).

%C 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]

%C Primes that occur in column 2 of A282293. - _Felix Fröhlich_, Aug 05 2018

%H R. Ernvall and T. Metsänkylä, <a href="http://dx.doi.org/10.1090/S0025-5718-97-00843-0">On the p-divisibility of Fermat quotients</a>, Math. Comp., Volume 66, Number 219 (1997), 1353-1365.

%H B. Logan and M. J. Mossinghoff, <a href="https://www.researchgate.net/publication/281628524">Double Wieferich pairs and circulant Hadamard matrices</a>, ResearchGate, 2015.

%t 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 *)

%o (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}))))

%Y Cf. A124122, A282293, A317724 (smallest existing q).

%K nonn,hard,more

%O 1,1

%A _Felix Fröhlich_, Jan 04 2016

%E a(5)-a(6) from _Felix Fröhlich_, May 29 2016

%E Removed three comments. - _Felix Fröhlich_, Aug 21 2018

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Last modified July 26 08:42 EDT 2021. Contains 346294 sequences. (Running on oeis4.)