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A076337 Riesel numbers: odd numbers n such that for all k >= 1 the numbers n*2^k - 1 are composite. 26
509203 (list; refs; listen; history; text; internal format)
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).
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.
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
Named after the Swedish mathematician Hans Ivar Riesel (1929-2014). - Amiram Eldar, Apr 02 2022
Paulo Ribenboim, The Book of Prime Number Records, 2nd ed., 1989, p. 282.
Ray Ballinger and Wilfrid Keller, The Riesel Problem: Definition and Status.
Chris Caldwell, Riesel Numbers.
Chris Caldwell, Sierpinski Numbers.
Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.9.8.
Tanya Khovanova, Non Recursions.
Joe McLean, Brier Numbers.
Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Riesel numbers.
Sequence in context: A205167 A252776 A271583 * A258154 A101036 A244070
N. J. A. Sloane, Nov 07 2002
Normally I require at least four terms but I am making an exception for this one in view of its importance. - N. J. A. Sloane, Nov 07 2002. See A101036 for the most likely extension.
Edited by N. J. A. Sloane, Nov 13 2009
Definition corrected ("odd" added) by M. F. Hasler, Aug 23 2020

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