# [seqfan] Re: Next conjecture help needed!

Max Alekseyev maxale at gmail.com
Sat Nov 22 19:03:14 CET 2008

```Artur,

I believe Mersenne primes do not satisfy your conjecture.
But otherwise notice that for 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.
Suppose that p is not Mersenne prime, implying that p is strictly
smaller than 2^m - 1. Let q>1 be a divisor of (2^m - 1)/p.
Then for any integer n such that p divides (2^n - 1), we have m
divides n, and thus (2^m - 1) divides (2^n - 1), implying that q
divides 2^n - 1 as well.
Therefore, p and q will always divide or not divide any 2^n - 1 together.

For Fermat numbers situation is different as any two distinct Fermat
numbers are co-prime. Therefore, divisor on on Fermat number is never
a divisor of another.

Regards,
Max

On Sat, Nov 22, 2008 at 9:51 AM, Artur <grafix at csl.pl> wrote:
> Dear Seqfans,
> Great thanks for Martin Fuller that back my esperance in my conjecture (my
> and from these reason my result 27 was with 2 small factors lack).
>
> My next conjencture is that some primes if occured as factors for Mersenne
> or Fermat numbers (mayby both) occured every time with partners and never
> separately e.g.
> 2^823-1
> let factors of number
> 24958107214398915181083907309638936320164123305586380092205774971508852\
>
> 912990276793728120564382191958225049401496767610430833092371951726531342764409\
> 228703366502075032590026785436867464508183220449
> will be a and b
>
> my conjecture say that if a is factor any number 2^x-1 and x>823 in this
> case also b is factor of these same number and vice versa if b is factor any
> number 2^x-1 and x>823 also a is is factor of these same number.
>
> Who can proove that or find contersample ?
>
> Best wishes
>
> Artur
>
> P.S.
>
> Oliver Gerard know better as yourself which topic or message is interesting
> post for you or not. And e.g. quick factorization of Mersenne or Fermat
> numbers is completely not interesting topic for all seqfans members and from
> these reason don't permit send any email to members of seqfans group. He
> know very well that anyone spam message will be more interesting.
>
>
> {{823,
> 24958107214398915181083907309638936320164123305586380092205774971508852\
> 912990276793728120564382191958225049401496767610430833092371951726531342764409\
> 228703366502075032590026785436867464508183220449}, {827, \
> 195542969191718936323989325793777292086070028324715108930713697626513616941626\
> 033838139352854980139297258251251040370115310454451037013620668240110688820210\
> 2548805683977577}, {853, \
> 119700352378963933859578380896030371926112856452654219449680221144328181225391\
> 946030194471333971484154060738372516425615009241775044835115081431304258506496\
> 32976327801924278410366010164414568098394520335715039457}, {857, \
> 140143233788391662452429369261365739864066680468767165333931912209073478199688\
> 152051519397027604260864607687684941271578715966808412817141243323172823378716\
> 710673041576729696645630513166955025926346421218242895748542260959270764327149\
> 755079125954280480903}
>
>
>

```