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

1,1

Comments

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

Links

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.

Mathematica

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 *)

Prog

(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}))))

Crossrefs

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

Sequence in context: A331021 A270833 A273471 * A203858 A115192 A307220

Adjacent sequences: A266826 A266827 A266828 * A266830 A266831 A266832

Keyword

nonn,hard,more

Author

Felix Fröhlich, Jan 04 2016

Extensions

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

Removed three comments. - Felix Fröhlich, Aug 21 2018

Status

approved