A282293 -id:A282293 - 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.

1093, 4871, 18787, 318917, 1006003, 1645333507

Offset: 1

Felix Fröhlich, Jan 04 2016

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

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.

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

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 *)

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}))))

a(5)-a(6) from Felix Fröhlich, May 29 2016

Removed three comments. - Felix Fröhlich, Aug 21 2018

A317724 Smallest prime q < A266829(n) such that both A266829(n)^(q-1) == 1 (mod q^2) and q^(A266829(n)-1) == 1 (mod A266829(n)^2), i.e., smallest prime q less than A266829(n) such that q and A266829(n) form a double Wieferich prime pair.

Original entry on oeis.org

2, 83, 2903, 911, 3, 5

Offset: 1

Felix Fröhlich, Aug 05 2018

			a(2) = 83, because 83 is the smallest prime q such that A266829(2) = 4871 satisfies both 4871^(q-1) == 1 (mod q^2) and q^(4871-1) == 1 (mod 4871^2).

Eric Weisstein's World of Mathematics, Double Wieferich Prime Pair

Cf. A266829. Supersequence of A124121.

Cf. A282293.

forprime(p=3, , forprime(q=2, p-1, if(Mod(p, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1, print1(q, ", "); break)))

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.

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