# [seqfan] Re: Next conjecture help needed!

Artur grafix at csl.pl
Sat Nov 22 19:53:53 CET 2008

```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)

Best wishes
Artur

Max Alekseyev pisze:
> 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}
>>
>>
>>
>>
>
> __________ Information from ESET NOD32 Antivirus, version of virus signature database 3632 (20081121) __________
>
> The message was checked by ESET NOD32 Antivirus.
>
> http://www.eset.com
>
>
>
>

```