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A076337 Riesel numbers: 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.


P. Ribenboim, The Book of Prime Number Records, 2nd. ed., 1989, p. 282.


Table of n, a(n) for n=1..1.

R. Ballinger and W. Keller, The Riesel Problem: Definition and Status

Chris Caldwell, Riesel Numbers

Chris Caldwell, Sierpinski Numbers

Yves Gallot, A search for some small Brier numbers, 2000.

Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, 16 (2013), #13.9.8.

Tanya Khovanova, Non Recursions

Joe McLean, Brier Numbers

C. Rivera, Brier numbers

Eric Weisstein's World of Mathematics, Riesel numbers


(MAGMA) P:=[3, 5, 7, 13, 17, 241]; C:=[0, 1, 0, 7, 3, 23]; Ch:=CRT([Modexp(2, C[i], P[i]): i in [1..#C]], P); R:=Ch/2^Valuation(Ch, 2); R; // Arkadiusz Wesolowski, Dec 09 2014


Cf. A076336, A076335, A003261, A052333, A101036.

Sequence in context: A124945 A205167 A252776 * A258154 A101036 A244070

Adjacent sequences:  A076334 A076335 A076336 * A076338 A076339 A076340




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



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Last modified December 1 15:12 EST 2015. Contains 264695 sequences.