

A266829


Primes p such that a prime q < p exists with p^(q1) == 1 (mod q^2) and q^(p1) == 1 (mod p^2), i.e., primes that are the larger member of a double Wieferich prime pair.


5




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 pdivisibility of Fermat quotients, Math. Comp., Volume 66, Number 219 (1997), 13531365.
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, p1, if(Mod(p, q^2)^(q1)==1 && Mod(q, p^2)^(p1)==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



