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Riesel Number

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There exist infinitely many odd integers k such that k·2^n-1 is composite for every n>=1. Numbers k with this property are called Riesel numbers, while analogous numbers with the minus sign replaced by a plus are called Sierpiński numbers of the second kind.

The smallest known Riesel number is k=509203, but as of Jan. 2014, there remain 52 smaller candidates which generate only composite numbers for all n which have been checked (Ribenboim 1996, p. 358; Ballinger and Keller; Keller): 2293, 9221, 23669, 31859, 38473, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 146561, 161669, 192971, 206039, 206231, 215443, 226153, 234343, 245561, 250027, 273809, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 402539, 409753, 444637, 470173, 474491, 477583, 485557, 494743, and 502573.

The problem of proving or disproving that k=509203 is the smallest Riesel number is sometimes known as the Riesel problem or Riesel conjecture.

Let a(k) be smallest n for which (2k-1)·2^n-1 is prime, then the first few values are 2, 0, 2, 1, 1, 2, 3, 1, 2, 1, 1, 4, 3, 1, 4, 1, 2, 2, 1, 3, 2, 7, ... (OEIS A046069), and second smallest n are 3, 1, 4, 5, 3, 26, 7, 2, 4, 3, 2, 6, 9, 2, 16, 5, 3, 6, 2553, ... (OEIS A046070).

Primes of the form k·2^n-1 discovered to date providing disproof of the existence of smaller Riesel numbers are summarized in the following table (Keller).

knDiscovererDate
659800516Dave Linton01 Mar 2004
267732465343Anonymous & RSP01 Dec 2006
27253272347Ray Ballinger10 Oct 1998
39269287048Richard Heylen25 Mar 2002
405976808509Frank Meador25 Dec 2013
42779322908Ray Ballinger26 Jul 1999
43541507098Ray Ballinger01 Oct 2000
46271428210Patrick Pirson29 Apr 2001
655313629342Adrian Schori & PrimeGrid05 Apr 2011
710091185112Drew Bishop & RSP05 Dec 2004
89707578313Richard Heylen02 Apr 2003
93997864401Guido Stolz & RSP01 Apr 2004
98939575144Olivier Haeberlé30 Nov 2001
103259615076Olivier Haeberlé23 Dec 2002
104917340181Janusz Szmidt13 Nov 1999
109897630221Olivier Haeberlé22 Apr 2003
1104131591999Will Fisher & RSP08 Jun 2005
1139833201175Ian Keogh & RSP01 May 2008
1144872198389Bruce White & RSP23 May 2006
1235473804809Jakub Łuszczek & PrimeGrid08 May 2011
126667626497Ray Ballinger09 Jun 2003
130139280296Dale Andrews02 Feb 2002
1419414299438Scott Brown & PrimeGrid26 May 2011
144643498079Richard Heylen12 Dec 2000
148901360338Mark Rodenkirch05 Mar 2002
1497971414137Peter van Hoof & RSP13 Mar 2005
1508471076441Darren Wallace & RSP15 Aug 2004
1527131154707Ray Ballinger23 Oct 2004
159371284166Janusz Szmidt14 Jan 2002
170591866870Drew Bishop & RSP15 Apr 2004
189463324103Dave Linton15 Jul 2000
1912493417696Jonathan Pritchard & PrimeGrid21 Nov 2010
1920891395688Guido Stolz & RSP10 May 2004
1965972178109Auritania Du & RSP09 May 2006
201193457615Daval Davis03 Feb 2003
204223696891Olivier Haeberlé23 Mar 2003
212893730387Olivier Haeberlé15 Oct 2003
215503649891Olivier Haeberlé28 Apr 2003
220033719731Olivier Haeberlé19 Apr 2004
220063306335Olivier Haeberlé03 Sep 1999
222997613153Olivier Haeberlé28 Nov 2001
2348471535589Darren Wallace & RSP09 May 2005
235601295338Helmut Zeisel06 Mar 2003
245051285750Tom Kuechler15 Nov 2000
246299752600Kevin O'Hare & RSP23 Jan 2004
2521915497878Jan Haller & PrimeGrid23 Jun 2012
261221689422Sean Faith & RSP22 Dec 2003
267763264115Dave Linton19 Feb 2000
2752932335007Japke Rosink & RSP21 Sep 2006
277153429819Jeff Wolfe21 Nov 2002
279703616235Dhumil Zaveri & RSP07 Jan 2004
299617428917Dave Linton22 Jul 2002
3042076643565Randy Ready & PrimeGrid11 Oct 2013
309817901173Helmut Michel & RSP07 Jun 2004
3256271472117Will Fisher & RSP05 Apr 2005
3426732639439Dhumil Zaveri & RSP28 Apr 2007
3450671876573Dave Linton13 Nov 2005
3501071144101Sean Faith & RSP24 Oct 2004
3531594331116Jaakko Reinman & PrimeGrid31 May 2011
357491609338Lucas Schmid17 Jan 2003
3576591779748Drew Bishop & RSP25 Sep 2005
376993293603Reto Keiser08 Sep 2002
382691431722Ray Ballinger27 Feb 2003
3980236418059Vladimir Volynsky & PrimeGrid05 Oct 2013
398533419107Dave Linton04 Sep 2002
401143532927Olivier Haeberlé11 Jun 2003
401617470149Dave Linton27 Dec 2002
4127171084409Holger Meissner & RSP22 Aug 2004
4152673771929Alexey Tarasov & PrimeGrid08 May 2011
416413424791Dave Linton28 Apr 2003
4176431800787Greg Childers & RSP05 Oct 2004
4286393506452Brett Melvold & PrimeGrid14 Jan 2011
443857369457Nuutti Kuosa27 Aug 2001
4504572307905Jeff Smith & RSP28 Mar 2006
458743547791Olivier Haeberlé22 Oct 2003
460139779536Drew Bishop & RSP26 Mar 2004
465869497596Lucas Schmid27 Jan 2003
4679171993429Steven Wong & RSP25 Dec 2005
4699491649228Steven Wong & RSP28 Oct 2007
4857673609357Chris Cardall & RSP24 Jun 2008
5006211138518Darren Wallace & RSP18 Oct 2004
5025411199930Ryan Sefko & RSP21 Dec 2004
5046131136459Magnus Mischel & RSP17 Oct 2004

See also

Brier Number, Cunningham Number, Mersenne Number, Sierpiński's Composite Number Theorem, Sierpiński Number of the Second Kind, Thâbit ibn Kurrah Rule

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References

Ballinger, R. and Keller, W. "The Riesel Problem: Search for Remaining Candidates." http://www.prothsearch.net/rieselsearch.html.Keller, W. "The Riesel Problem: Definition and Status." http://www.prothsearch.net/rieselprob.html.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 357, 1996.Riesel, H. "Några stora primtal." Elementa 39, 258-260, 1956.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkhäuser, pp. 394-398, 1994.Riesel Sieve Project. "The Riesel Sieve Project: A Distributed Effort to Prove the Riesel Conjecture." http://www.rieselsieve.com/.Sloane, N. J. A. Sequences A046067, A046068, A046069, and A046070 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Riesel Number

Cite this as:

Weisstein, Eric W. "Riesel Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RieselNumber.html

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