# [seqfan] Re: Problem

Allan Wechsler acwacw at gmail.com
Sat Jul 25 00:01:03 CEST 2020

Dzień dobry, Tomasz. To clarify, I absolutely agree that the theorem you
stated is true, and I had a very similar proof in my head. The thing that
is very hard to prove is the "left hand side" of the theorem, that for all
primes p there is a larger prime of the form (p+1)2^k - 1. I don't even
have a strong intuition that it is true. If it WERE true, then we could use
it to establish once and for all that there are an infinite number of
Mersenne primes (and hence perfect numbers).

Alarming examples like W. Edwin Clark's 94603 cast doubt in my mind.

On Fri, Jul 24, 2020 at 5:14 PM W. Edwin Clark <wclark at mail.usf.edu> wrote:

> For the prime p = 94603 and for n from 1 to 100000, (p+1) 2^n - 1 is
> composite, says Maple.
> This prime appears twice in the OEIS if you don't count A094603.  See
> http://oeis.org/search?q=94603&language=english&go=Search   Note this
> search doesn't
> include things like the sequence of primes.
>
> On Sun, Jul 19, 2020 at 2:28 AM Tomasz Ordowski <tomaszordowski at gmail.com>
> wrote:
>
> > Dear SeqFans!
> >
> > Let a(0) = p and a(n) = 2 a(n-1) + 1. Note that a(n) = (p+1) 2^n - 1.
> > Are there primes p such that a(n) is composite for every n > 0 ?
> >
> > Best regards,
> >
> > Thomas Ordowski
> > _______________________
> > https://en.wikipedia.org/wiki/Riesel_number
> >
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> >
>
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