Compiled by Wilfrid Keller
In 1956 Hans Riesel proved the following interesting result.
Theorem. There exist infinitely many odd integers k such that k · 2^{n} − 1 is composite for every n > 1.
Actually, Riesel showed that k_{0} = 509203 has this property, and also the multipliers k_{r} = k_{0} + 11184810r for r = 1, 2, 3, . . . Such numbers are now called Riesel numbers because of their similarity with the Sierpiński numbers. Note that k_{0} is a prime number. The Riesel problem consists in determining the smallest Riesel number.
Conjecture. The integer k = 509203 is the smallest Riesel number.
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, 502573 
The latest update to the list is from December 28, 2013.
A reasonable approach to the Riesel problem is to determine the first exponent n giving a prime k · 2^{n} − 1 in each case. So we can observe the exact rate at which the 254601 multipliers k < 509203 are successively eliminated, which may enable us to predict their further decrease by extrapolation.


k  n  Discoverer  Date 
27253  272347  Ray Ballinger  10 Oct 1998 
39269  287048  Richard Heylen  25 Mar 2002 
42779  322908  Ray Ballinger  26 Jul 1999 
43541  507098  Ray Ballinger  01 Oct 2000 
46271  428210  Patrick Pirson  29 Apr 2001 
104917  340181  Janusz Szmidt  13 Nov 1999 
130139  280296  Dale Andrews  02 Feb 2002 
144643  498079  Richard Heylen  12 Dec 2000 
148901  360338  Mark Rodenkirch  05 Mar 2002 
159371  284166  Janusz Szmidt  14 Jan 2002 
189463  324103  Dave Linton  15 Jul 2000 
201193  457615  Daval Davis  03 Feb 2003 
220063  306335  Olivier Haeberlé  03 Sep 1999 
235601  295338  Helmut Zeisel  06 Mar 2003 
245051  285750  Tom Kuechler  15 Nov 2000 
267763  264115  Dave Linton  19 Feb 2000 
277153  429819  Jeff Wolfe  21 Nov 2002 
299617  428917  Dave Linton  22 Jul 2002 
376993  293603  Reto Keiser  08 Sep 2002 
382691  431722  Ray Ballinger  27 Feb 2003 
398533  419107  Dave Linton  04 Sep 2002 
401617  470149  Dave Linton  27 Dec 2002 
416413  424791  Dave Linton  28 Apr 2003 
443857  369457  Nuutti Kuosa  27 Aug 2001 
465869  497596  Lucas Schmid  27 Jan 2003 
k  n  Discoverer  Date 
659  800516  Dave Linton  01 Mar 2004 
89707  578313  Richard Heylen  02 Apr 2003 
93997  864401  Guido Stolz & RSP  01 Apr 2004 
98939  575144  Olivier Haeberlé  30 Nov 2001 
103259  615076  Olivier Haeberlé  23 Dec 2002 
109897  630221  Olivier Haeberlé  22 Apr 2003 
126667  626497  Ray Ballinger  09 Jun 2003 
170591  866870  Drew Bishop & RSP  15 Apr 2004 
204223  696891  Olivier Haeberlé  23 Mar 2003 
212893  730387  Olivier Haeberlé  15 Oct 2003 
215503  649891  Olivier Haeberlé  28 Apr 2003 
220033  719731  Olivier Haeberlé  19 Apr 2004 
222997  613153  Olivier Haeberlé  28 Nov 2001 
246299  752600  Kevin O'Hare & RSP  23 Jan 2004 
261221  689422  Sean Faith & RSP  22 Dec 2003 
279703  616235  Dhumil Zaveri & RSP  07 Jan 2004 
309817  901173  Helmut Michel & RSP  07 Jun 2004 
357491  609338  Lucas Schmid  17 Jan 2003 
401143  532927  Olivier Haeberlé  11 Jun 2003 
458743  547791  Olivier Haeberlé  22 Oct 2003 
460139  779536  Drew Bishop & RSP  26 Mar 2004 
k  n  Discoverer  Date 
162941  993718  Dmitry Domanov & PrimeGrid  02 Feb 2012 
k  n  Discoverer  Date 
71009  1185112  Drew Bishop & RSP  05 Dec 2004 
110413  1591999  Will Fisher & RSP  08 Jun 2005 
149797  1414137  Peter van Hoof & RSP  13 Mar 2005 
150847  1076441  Darren Wallace & RSP  15 Aug 2004 
152713  1154707  Ray Ballinger  23 Oct 2004 
192089  1395688  Guido Stolz & RSP  10 May 2004 
234847  1535589  Darren Wallace & RSP  09 May 2005 
325627  1472117  Will Fisher & RSP  05 Apr 2005 
345067  1876573  Dave Linton  13 Nov 2005 
350107  1144101  Sean Faith & RSP  24 Oct 2004 
357659  1779748  Drew Bishop & RSP  25 Sep 2005 
412717  1084409  Holger Meissner & RSP  22 Aug 2004 
417643  1800787  Greg Childers & RSP  05 Oct 2004 
467917  1993429  Steven Wong & RSP  25 Dec 2005 
469949  1649228  Steven Wong & RSP  28 Oct 2007 
500621  1138518  Darren Wallace & RSP  18 Oct 2004 
502541  1199930  Ryan Sefko & RSP  21 Dec 2004 
504613  1136459  Magnus Mischel & RSP  17 Oct 2004 
As a combined result of the above computations (including the correction), 71 values of k were left which had no prime k · 2^{n}  1 for n < 2097152 = 2^{21}. From these 71 undecided values of k another 8 have been eliminated by the Riesel Sieve Project, whose participants discovered primes k · 2^{n} − 1 for the following pairs k, n :
k  n  Discoverer  Date 
26773  2465343  John Dalton & RSP  01 Dec 2006 
113983  3201175  Ian Keogh & RSP  01 May 2008 
114487  2198389  Bruce White & RSP  23 May 2006 
196597  2178109  Auritania Du & RSP  09 May 2006 
275293  2335007  Japke Rosink & RSP  21 Sep 2006 
342673  2639439  Dhumil Zaveri & RSP  28 Apr 2007 
450457  2307905  Jeff Smith & RSP  28 Mar 2006 
485767  3609357  Chris Cardall & RSP  24 Jun 2008 
Following the unfortunate disappearence of the Riesel Sieve Project, the investigation was finally resumed at PrimeGrid, starting in March 2010: see The Riesel Problem. In this context, another 5 primes k · 2^{n} − 1 having 2097152 ≤ n < 4194304 were discovered, as follows; the values of k are linked to PrimeGrid's Official Announcements:
k  n  Discoverer  Date 
65531  3629342  Adrian Schori & PrimeGrid  05 Apr 2011 
123547  3804809  Jakub Łuszczek & PrimeGrid  08 May 2011 
191249  3417696  Jonathan Pritchard & PrimeGrid  21 Nov 2010 
415267  3771929  Alexey Tarasov & PrimeGrid  08 May 2011 
428639  3506452  Brett Melvold & PrimeGrid  14 Jan 2011 
On June 4, 2011 the search was completed to the limit n < 4194304 = 2^{22} for all of the remaining 58 "Riesel number candidates", which finally established the value of f_{21} = 13 (see table of frequencies above). The search is continuing for n ≥ 4194304 and has currently covered all n < 6783000. The following additional primes have been found so far:
k  n  Discoverer  Date 
40597  6808509  Frank Meador & PrimeGrid  25 Dec 2013 
141941  4299438  Scott Brown & PrimeGrid  26 May 2011 
252191  5497878  Jan Haller & PrimeGrid  23 Jun 2012 
304207  6643565  Randy Ready & PrimeGrid  11 Oct 2013 
353159  4331116  Jaakko Reinman & PrimeGrid  31 May 2011 
398023  6418059  Vladimir Volynsky & PrimeGrid  05 Oct 2013 
Overall, this leaves the 52 uncertain candidates listed at the
beginning. The largest prime discovered during this investigation is the 2049571
digit
prime 40597 · 2^{6808509} − 1.
References.
For more information see the Riesel number page in Chris Caldwell's Glossary.
Address questions about this web page to Wilfrid Keller