A253683 Primes p in increasing order with p > A253684(n) > A253685(n) such that (p, A253684(n), A253685(n)) forms a Wieferich triple.

71, 863, 1093, 2281, 3511, 13691, 20771, 54787, 950507, 1843757, 3188089

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) > 121637. - Felix Fröhlich, Jun 18 2016

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

Links

• Wikipedia, Wieferich pair

Crossrefs

Cf. A124121, A124122, A253684, A253685.

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(p, ", ")))))

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