# [seqfan] Re: An unknown problem

Tomasz Ordowski tomaszordowski at gmail.com
Sun Oct 14 11:16:10 CEST 2018

```P.S. Supplement:

Cf. https://oeis.org/A298076 and https://oeis.org/A298310.

Thomas Ordowski

pon., 1 paź 2018 o 07:46 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear SeqFans!
>
> The problem:
>
> Are there composite numbers n such that 2^n - 1 has all prime divisors p
> == 1 (mod n) ?
>
>   Note: Such a supposed number n must be a Fermat pseudoprime to base 2.
>
> If both 2^p - 1 and 2^q - 1 are prime and n = pq is pseudoprime, then n is
> such a number.
>
> However, known Mersenne primes do not give such pseudoprimes.
>
> Best regards,
>
> Thomas
> ______________
> There are similar composite numbers n to other bases,
> for example 91 = 7*13 to base 3 and 341 = 11*31 to base -2;
> the number (3^91-1)/2 has all prime divisors p == 1 (mod 91)
> and (2^341+1)/3 has all prime divisors p == 1 (mod 341).
> Note that both (3^7-1)/2 and (3^13-1)/2 are prime
> and both (2^11+1)/3 and (2^31+1)/3 are prime.
>
>

```