# [seqfan] Re: Problem

Tomasz Ordowski tomaszordowski at gmail.com
Mon Jul 20 19:00:32 CEST 2020

```Hello Robert,

Yes, the number (73+1)2^2552-1 is prime.

Thomas

P.S. Analogous problem goes beyond the Sierpinski numbers.
Let a(n) = 2 a(n-1) - 1 with a(0) = p. Note that a(n) = (p-1) 2^n + 1.
Are there primes p such that a(n) is composite for every n > 0 ?
It is known that there exist such composites:
even m = k+1 and odd 2m-1 = 2k+1,
where k is a Sierpinski number.
Maybe only p = 2^(2^4)+1.

pon., 20 lip 2020 o 18:30 Robert Dougherty-Bliss <robert.w.bliss at gmail.com>
napisał(a):

> Dear Thomas,
>
> You may already be aware, but none of the first 100 primes (<= 541)
> satisfy this property.
>
> Amazingly, the earliest counterexample for p = 73 is the following integer:
>
>
> 12525084203259602214176345117827991857573063437151079650189656689252041617399
>
> 16118618976873174436648194378202145606096817433350319763375794132326993383200
>
> 14217732225003163760036417965916387747831867749318699104524437655151695087826
>
> 47278357731824391729532319069188907350539418959168425940169356532195426353195
>
> 84257183520755212129194474630919879413057346247800071524008686049488780942766
>
> 38123436651683349651892026768245860789398297612527549211852109219078820059778
>
> 19346432242814374609091413789240598598335924463948419947004368457022517766034
>
> 95591799870311650343246943884972083691195975663585667560716289785503524182355
>
> 53897768571561351251352502155056787443177087759615376430034900988921205572639
>
> 317118528079725593399200244440233458975807425711011346463660588817113315016703
>
> Robert
>
>
> Robert
>
>
> 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
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```