[seqfan] Re: Glued primes

Max Alekseyev maxale at gmail.com
Sun Nov 23 18:41:18 CET 2008

On Sun, Nov 23, 2008 at 8:49 AM, Artur <grafix at csl.pl> wrote:
> Dear Max,
> I was check few samples
> e.g. pair of primes {10429407431911334611, 918125051602568899753}
> which are two the biggest prime divisors of 2^243+1 and 2^486-1
> 486=2*243
> occured every time in pairs (never separately) and both in Fermat as in
> Mersenne numbers (infinite many times)

You don't need to check that manually. For any number n that is
multiple of 243,
the number 2^n - 1 is divisible by both these primes. Similarly, for
an odd multiple m of 243, the number 2^m + 1 is divisible by these
primes as well.

btw, usually Fermat numbers are those of the form 2^(2^n) + 1 not just
2^m + 1 - see http://en.wikipedia.org/wiki/Fermat_number
It's very confusing when you call just arbitrary number of the form
2^m + 1 a Fermat number.

> (I know few more such pairs) and many many products of such pairs but I
> can't factorized these on separate primes because I was mentioned yesterday
> these cases are most difficult to factorization in anyone method.
> I don't know yet that triples and more primes product occured but I'm
> examined these now.

I suggest a triple [233,1103,2089] which are the prime factors of 2^29
- 1, all three of the multiplicative order 29.
They always together divide or not divide Mersenne numbers 2^n - 1 as
well as numbers of the form 2^m + 1.


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