# [seqfan] Re: Next conjecture help needed!

Artur grafix at csl.pl
Sat Nov 22 20:21:44 CET 2008

```Dear Max,
I was understand now, you only proove that doesn't exist such q when p
is Mersenne prime and nothing else.
But of course any Mersenne prime don't divide any composite number which
divisors are candidates to my conjecture.
That we are in start point.
Best wishes
Artur

Max Alekseyev pisze:
> 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
>
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```