cp's OEIS Frontend

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.

%I A266829 #49 Aug 21 2018 04:21:02
%S A266829 1093,4871,18787,318917,1006003,1645333507
%N 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.
%C A266829 There are no further terms less than 10^6 (cf. Ernvall, Metsänkylä, 1997, p. 1360).
%C A266829 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 A266829 Primes that occur in column 2 of A282293. - _Felix Fröhlich_, Aug 05 2018
%H A266829 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 A266829 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 A266829 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 A266829 (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 A266829 Cf. A124122, A282293, A317724 (smallest existing q).
%K A266829 nonn,hard,more
%O A266829 1,1
%A A266829 _Felix Fröhlich_, Jan 04 2016
%E A266829 a(5)-a(6) from _Felix Fröhlich_, May 29 2016
%E A266829 Removed three comments. - _Felix Fröhlich_, Aug 21 2018