[seqfan] An unknown problem

[Tomasz Ordowski]

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.