Compiled by Wilfrid Keller
In 1956 Hans Riesel (1929−2014) proved the following interesting result.
Theorem. There exist infinitely many odd integers k such that k · 2n − 1 is composite for every n > 1.
Actually, Riesel showed that k0 = 509203 has this property, and also the multipliers kr = k0 + 11184810r for r = 1, 2, 3, . . . Such numbers are now called Riesel numbers because of their similarity with the Sierpiński numbers. Note that k0 is a prime number. The Riesel problem consists in determining the smallest Riesel number.
Conjecture. The integer k = 509203 is the smallest Riesel number.
23669, 31859, 38473, 46663, 67117, 74699, 81041, 107347, 121889, 129007, | 143047, 161669, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, | 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, | 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, | 485557, 494743 |
The latest update to the list is from May 1st, 2023.
A reasonable approach to the Riesel problem is to determine the first exponent n giving a prime k · 2n − 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.
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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 · 2n - 1 for n < 2097152 = 221. From these 71 undecided values of k another 8 have been eliminated by the Riesel Sieve Project, whose participants discovered primes k · 2n − 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 · 2n − 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 |
with the count of f22 = 8 (compare the table of
frequencies above).
However, five even larger primes were subsequently reported by an independent researcher:
In these cases it is not known whether the given primes are the smallest ones in their respective sequences,
as the search was temporarily stopped near n = 11534000, 13977000, 14450000, 11534000, and 11629000,
respectively. For all the other 42 uncertain candidates listed at the beginning,
the coordinated search has currently reached n < 14447000. Under these circumstances, the last two
entries in the table of frequencies should be regarded with caution.
Obviously, the largest prime discovered during this investigation is the
recent 5538219-digit prime 97139 · 218397548 − 1.
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
402539
7173024
Walter Darimont & PrimeGrid
02 Oct 2014
502573
7181987
Denis Iakovlev & PrimeGrid
04 Oct 2014
k
n
Discoverer
Date
9221
11392194
Barry Schnur & PrimeGrid
07 Feb 2021
146561
11280802
Pavel Atnashev & PrimeGrid
16 Nov 2020
273809
8932416
Wolfgang Schwieger & PrimeGrid
13 Dec 2017
k
n
Discoverer
Date
2293
12918431
Ryan Propper
13 Feb 2021
93839
15337656
Ryan Propper
27 Nov 2022
97139
18397548
Ryan Propper
23 Apr 2023
192971
14773498
Ryan Propper
07 Mar 2021
206039
13104952
Ryan Propper
26 Apr 2021
References.
For more information see the Riesel number page in Chris Caldwell's Glossary.
Please address questions about this web page to Wilfrid Keller