# Riesel numbers and others

Richard Guy rkg at cpsc.ucalgary.ca
Wed Mar 5 19:35:05 CET 2003

```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
> 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

```