Riesel numbers and others

[David Wilson]

Here is the difficulty with Riesel and Sierpinsky numbers.  I will argue
using Riesel numbers, the argument is exactly analogous for the similar
Sierpinski numbers.

Let R_n(k) = n*2^k+1.  A number n is proved non-Riesel by exhibiting a
prime R_n(k).  In all proofs I know of, a number n is proven Riesel by
exhibiting a periodic sequence of proper primes p_n with p_n(k) a proper
divisor of R_n(k), which precludes a prime R_n(k).

The smallest number proved Riesel is 509203.  However, there are
several smaller numbers n whose status is not known.  No periodic
sequence p_n of proper divisors has been found for R_n, nor has a prime
been found in R_n, despite extensive search.  These numbers remain
candidate Riesel numbers.  I suppose there must be probabilistic argument
that is p_n does not exist, n is probably non-Riesel, however, this does
not count as a proof. I also do not know how the existence of p_n is
ruled out for any particular n.

As far as sequences go, we might make sequences of proved Riesel
numbers (numbers n for which p_n has been found) and/or candidate
Riesel numbers (numbers n for which neither p_n nor prime R_n(k)
have yet been found).  These sequences would of course be subject
to update as new data become available.

A076336 and A076337 below are obviously attempts as sequences of
proved Riesel and Sierpinski numbers (actually, the initial elements are
mixed up in the A076336 and A076337 below, but have been fixed in the
database).

----- Original Message -----
From: "Richard Guy" <rkg at cpsc.ucalgary.ca>
To: "N. J. A. Sloane" <njas at research.att.com>
Cc: <seqfan at ext.jussieu.fr>
Sent: Wednesday, March 05, 2003 1:35 PM
Subject: Re: Riesel numbers and others


> Slightly more seriously, according to
> UPINT B21, Sierpi'nski showed that
> k*2^n + 1  was composite for all  n
> if  k = 1 mod 641*(2^32 - 1)
>   and  -1 mod 6700417
> This gives an infinity of members, but
> presumably you can't include them, 'cos
> there are smaller ones.  Add as a remark??
>     R.
>
> On Thu, 7 Nov 2002, N. J. A. Sloane wrote:
>
> > A recent email from Olivier led me to
> > add these three sequences.  The 2nd and 3rd badly need
> > extending!    Neil
> >
> > %I A076335
> > %S A076335 878503122374924101526292469,3872639446526560168555701047,
> > %T A076335 623506356601958507977841221247
> > %N A076335 Brier numbers: both Riesel and Sierpinski, or n such that for all k >= 0 the numbers
n*2^k + 1 and n*2^k - 1 are composite.
> > %C A076335 These are just the smallest examples known - there may be smaller ones.
> > %Y A076335 Cf. A076336, A076337.
> > %H A076335 Yves Gallot, <a href="http://perso.wanadoo.fr/yves.gallot/papers/smallbrier.html">A
search for some small Brier numbers</a>, 2000.
> > %H A076335 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_029.htm">Brier
numbers</a>
> > %K A076335 nonn,new
> > %O A076335 1,1
> > %A A076335 Olivier Gerard (ogerard at ext.jussieu.fr), Nov 07 2002
> >
> >
> > %I A076336
> > %S A076336 78557
> > %N A076336 Riesel numbers: n such that for all k >= 0 the numbers n*2^k + 1 are composite.
> > %Y A076336 Cf. A076337, A076335, A003261.
> > %K A076336 nonn,new,bref,hard,more
> > %H A076336 Yves Gallot, <a href="http://perso.wanadoo.fr/yves.gallot/papers/smallbrier.html">A
search for some small Brier numbers</a>, 2000.
> > %H A076336 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_029.htm">Brier
numbers</a>
> > %O A076336 1,1
> > %A A076336 njas, Nov 07 2002
> > %E A076336 Normally I require at least four terms but I am making an exception for this one in
the hope that someone will extend it. - njas, Nov 07, 2002.
> >
> >
> > %I A076337
> > %S A076337 509203
> > %N A076337 Sierpinski numbers: n such that for all k >= 0 the numbers n*2^k - 1 are composite.
> > %Y A076337 Cf. A076337, A076335.
> > %K A076337 nonn,new,bref,hard,more
> > %H A076337 Yves Gallot, <a href="http://perso.wanadoo.fr/yves.gallot/papers/smallbrier.html">A
search for some small Brier numbers</a>, 2000.
> > %H A076337 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_029.htm">Brier
numbers</a>
> > %O A076337 1,1
> > %A A076337 njas, Nov 07 2002
> > %E A076337 Normally I require at least four terms but I am making an exception for this one in
the hope that someone will extend it. - njas, Nov 07, 2002.
> >
> > Neil Sloane
>
>