A253685 Primes r with A253683(n) > A253684(n) > r such that (A253683(n), A253684(n), r) is a Wieferich triple.

3, 13, 2, 1657, 2, 83, 5, 431, 5, 199, 3

Offset: 1

Author

Felix Fröhlich, Jan 09 2015

Keywords

• nonn
• hard
• more

Comments

In analogy to a Wieferich pair, a set of three primes p, q, r can be called a 'Wieferich triple' if its members satisfy either of the following two sets of congruences:

p^(q-1) == 1 (mod q^2), q^(r-1) == 1 (mod r^2), r^(p-1) == 1 (mod p^2)

p^(r-1) == 1 (mod r^2), r^(q-1) == 1 (mod q^2), q^(p-1) == 1 (mod p^2)

a(9) must have A253683(n) > 121637. - Felix Fröhlich, Jun 18 2016

a(12) must have A253683(n) > 5*10^6. - Giovanni Resta, Jun 20 2016

Links

• Wikipedia, Wieferich pair

Crossrefs

Cf. A124121, A124122.

Cf. A253683, A253684.

Programs

PARI

forprime(p=1, , forprime(q=1, p, forprime(r=1, q, if((Mod(p, q^2)^(q-1)==1 && Mod(q, r^2)^(r-1)==1 && Mod(r, p^2)^(p-1)==1) || (Mod(p, r^2)^(r-1)==1 && Mod(r, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1), print1(r, ", ")))))

Extensions

a(8) from Felix Fröhlich, Jun 18 2016

Name edited by Felix Fröhlich, Jun 18 2016

a(9)-a(11) from Giovanni Resta, Jun 20 2016