2*k*p+1.Re: prime sequences A080050/A080051

[Don McDonald]

In message <20030125121629.A839 at ark.in-berlin.de> you write:
Any prime divisor of 2^prime -1
can only be of the form  2*k*p +1.

I have heard this from David M Burton book
Elementary (theory of numbers)
and Bob Silverman.

E.g. Reader guide abstracts mistakenly reported
world's largest known prime number at one time = 2^858433-1.
Which I can eliminate.

(Has a factor 70*p+1 ?)
2^859433-1 was the (correct) Mersenne prime.

I can eliminate many exponents of the order of world's
largest known prime (year 2001.)

Mersenne primes are binary prime repunits. 2^p-1.= 111.111.

But one has a very unusual palindrome (capicue) representation.

p = 1*2*34*44432 +1, table mountain.
         - ---
        /     \
      /        \
    /             \
    =3021377.

(Poster, World year of mathematics, 2000.
European mathematical soc.)
Donald S. McDonald.

don.mcdonald at paradise.net.nz
26.01.03  22:45


> Can someone prove there is a mapping between
> 
> %N A080050 Primes p such that 2^p-1 and the p-th Fibonacci number have a common
>  factor. Prime members of A074776.
> and
> %N A080051 Common factors of 2^p-1 and the p-th Fibonacci number, p prime.
> ?
> Define p the former, gcd the latter numbers.
> 
> gcd = 8k(p-1)+9 ???
> x is the position of the twin prime pair.
> All shown gcd are prime.
> 
> Thanks,
> ralf
> 
> p      gcd      estim.(gcd-9)/(8p) please verify
> -------------------
> 11     89       1
> 8501   680081   10
> 10867  260809   3
> 13109  3146161  30
> 14633  1170641  10
> 15401  123209   1
>
...                2*k*p+1. common factor.