%I #77 Sep 21 2024 12:36:55

%S 509203

%N Riesel numbers: odd numbers n such that for all k >= 1 the numbers n*2^k - 1 are composite.

%C 509203 has been proved to be a member of the sequence, and is conjectured to be the smallest member. However, as of 2009, there are still several smaller numbers which are candidates and have not yet been ruled out (see links).

%C Riesel numbers are proved by exhibiting a periodic sequence p of prime divisors with p(k) | n*2^k-1 and disproved by finding prime n*2^k-1. It is conjectured that numbers that cannot be proved Riesel in this way are non-Riesel. However, some numbers resist both proof and disproof.

%C Others conjecture the opposite: that there are infinitely many Riesel numbers that do not arise from a covering system, see A101036. The word "odd" is needed in the definition because otherwise for any term n, all numbers n*2^m, m >= 1, would also be Riesel numbers, but we don't want them in this sequence (as is manifest from A101036). Since 1 and 3 obviously are not in this sequence, for any n in this sequence n-1 is an even number > 2 and therefore composite, so one could replace "k >= 1" equivalently by "k >= 0". - _M. F. Hasler_, Aug 20 2020

%C Named after the Swedish mathematician Hans Ivar Riesel (1929-2014). - _Amiram Eldar_, Apr 02 2022

%D R. K. Guy, Unsolved Problems in Number Theory, Section B21.

%D Paulo Ribenboim, The Book of Prime Number Records, 2nd ed., 1989, p. 282.

%H Ray Ballinger and Wilfrid Keller, <a href="http://www.prothsearch.com/rieselprob.html">The Riesel Problem: Definition and Status</a> [http://www.prothsearch.com/rieselprob.html].

%H Chris Caldwell, <a href="https://t5k.org/glossary/page.php?sort=RieselNumber">Riesel Numbers</a>.

%H Chris Caldwell, <a href="https://t5k.org/glossary/page.php?sort=SierpinskiNumber">Sierpinski Numbers</a>.

%H Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf">A search for some small Brier numbers</a>, 2000.

%H Dan Ismailescu and Peter Seho Park, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.html">On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.9.8.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/NonRecursions.html">Non Recursions</a>.

%H Joe McLean, <a href="http://www.glasgowg43.freeserve.co.uk/brier2.htm">Brier Numbers</a>.

%H Hans Riesel, <a href="/A076337/a076337.pdf">Some large prime numbers</a>. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_029.htm">Problem 29. Brier numbers</a>, The Prime Puzzles and Problems Connection.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RieselNumber.html">Riesel numbers</a>.

%H <a href="/index/O#oneterm">Index entries for one-term sequences</a>.

%Y Main sequences for Riesel problem: A040081, A046069, A050412, A052333, A076337, A101036, A108129.

%Y Cf. A076336, A076335, A003261.

%K nonn,bref,hard,more

%O 1,1

%A _N. J. A. Sloane_, Nov 07 2002

%E Normally we require at least four terms but we will make an exception for this sequence in view of its importance. - _N. J. A. Sloane_, Nov 07 2002. See A101036 for the most likely extension.

%E Edited by _N. J. A. Sloane_, Nov 13 2009

%E Definition corrected ("odd" added) by _M. F. Hasler_, Aug 23 2020