# Riesel numbers and others

David Wilson davidwwilson at attbi.com
Thu Nov 7 22:35:21 CET 2002

```First off, I think NJAS has the definitions of Riesel and Sierpinsky numbers
mixed up.
See e.g:

http://mathworld.wolfram.com/RieselNumber.html
http://primes.utm.edu/glossary/page.php?sort=RieselNumber
http://www.glasgowg43.freeserve.co.uk/brier2.htm

There is also another problem with adding a sequence Riesel numbers.  It is
possible
to show a number n is non-Riesel by exhibiting a prime 2^k n - 1.  It is
possible to show
n is Riesel by exhibiting a cyclic sequence d of divisors with d_k | 2^k n +
1.  To my
knowledge, these are the only known methods of estabilishing with certainty
whether
or not a given n is Riesel.

It is conjectured that any number n for which the cyclic divisor sequence d
cannot be
established is non-Riesel.  I assume there is an argument that such numbers
are
non-Riesel with probability 1.  This does not however, prove that they are
non-Riesel.
The smallest number proved Riesel by cyclic divisors is 509203.  However
there are
still several number n < 509203 for which neither prime 2^k n - 1 nor cyclic
sequence d
with d_k | 2^k n + 1 has been found.  The status of these numbers is
unknown.

Therefore, any sequence of Riesel numbers must either (1) include numbers of
unknown
status, to be dropped as they shown non-Riesel, or (2) include only Riesel
numbers which
have been proved by exhibiting a cyclic divisor sequence.  However, I do not
know how
one eliminates the possibility of such a cyclic divisor sequence for a given
n.

----- Original Message -----
From: "N. J. A. Sloane" <njas at research.att.com>
To: <njas at research.att.com>; <seqfan at ext.jussieu.fr>
Sent: Thursday, November 07, 2002 11:18 AM
Subject: Riesel numbers and others

>
> A recent email from Olivier led me to
> 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
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
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
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
>

```