A229145 Numbers k such that (36^k + 1)/37 is prime.

31, 191, 257, 367, 3061, 110503, 1145393

Offset: 1

Author

Robert Price, Sep 15 2013

Keywords

• hard
• more
• nonn

Comments

All such numbers k are prime.

Note that a(6) = 110503 corresponds to (36^110503 + 1)/37, which is only a probable prime with 171975 digits.

The primes corresponding to the terms of this sequence have 1 as their last digit and an even number as their next-to-last digit. - Iain Fox, Dec 08 2017

Links

• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
• H. Lifchitz, Mersenne and Fermat primes field
• Eric Weisstein's World of Mathematics, Repunit.

Crossrefs

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240.

Programs

Mathematica

Do[ p=Prime[n]; If[ PrimeQ[ (36^p + 1)/37 ], Print[p] ], {n, 1, 9592} ]

PARI

is(n)=isprime((36^n+1)/37) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(6) = 110503 (posted by Lelio R. Paula on primenumbers.net) from Paul Bourdelais, Dec 08 2017

a(7) from Paul Bourdelais, Nov 03 2023