# [seqfan] Re: Next conjecture help needed!

Max Alekseyev maxale at gmail.com
Sat Nov 22 20:13:42 CET 2008

```On Sat, Nov 22, 2008 at 10:53 AM, Artur <grafix at csl.pl> wrote:
> Dear Max,
> 1)
>>I believe Mersenne primes
> I hope your  "Mersenne primes" you mean Mersenne numbers.
> 2)
>
>> an odd prime number p there exists a
>> smallest positive integer m (called the multiplicative order of 2
>
> <modulo p) such that p divides 2^m - 1.
>
> For every odd prime number p there exists a smallest positive integer m
> (called the multiplicative order of 2 modulo p) such that p divides 2^m - 1.
>
> but this not say that can't existed such q that also divided 2^m - 1 for
> that same value m (p and q have that same multiplicative order of 2 modulo p
> and q)

1) Saying "Mersenne prime" I meant Mersenne prime, not just number - see below:

2) Such q does not exist if p is a Mersenne prime, in which case p =
2^m - 1 and there is no any non-trivial co-factor of p in the
factorization of 2^m - 1.
For example, let p = 2^2 - 1 = 3 (a Mersenne prime).
Then p does not have any "mate" since, for example, 3 is the only
common factor of 2^4-1=3*5 and 2^6-1=63=3*3*7.

Regards,
Max

```