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There exist infinitely many odd integers such that is composite for every . Numbers 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 , but as of Jan. 2014, there remain 52 smaller candidates which generate only composite numbers for all 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 is the smallest Riesel number is sometimes known as the Riesel problem or Riesel conjecture.
Let be smallest for which 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 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 discovered to date providing disproof of the existence of smaller Riesel numbers are summarized in the following table (Keller).
Discoverer | Date | ||
659 | 800516 | Dave Linton | 01 Mar 2004 |
26773 | 2465343 | Anonymous & RSP | 01 Dec 2006 |
27253 | 272347 | Ray Ballinger | 10 Oct 1998 |
39269 | 287048 | Richard Heylen | 25 Mar 2002 |
40597 | 6808509 | Frank Meador | 25 Dec 2013 |
42779 | 322908 | Ray Ballinger | 26 Jul 1999 |
43541 | 507098 | Ray Ballinger | 01 Oct 2000 |
46271 | 428210 | Patrick Pirson | 29 Apr 2001 |
65531 | 3629342 | Adrian Schori & PrimeGrid | 05 Apr 2011 |
71009 | 1185112 | Drew Bishop & RSP | 05 Dec 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 |
104917 | 340181 | Janusz Szmidt | 13 Nov 1999 |
109897 | 630221 | Olivier Haeberlé | 22 Apr 2003 |
110413 | 1591999 | Will Fisher & RSP | 08 Jun 2005 |
113983 | 3201175 | Ian Keogh & RSP | 01 May 2008 |
114487 | 2198389 | Bruce White & RSP | 23 May 2006 |
123547 | 3804809 | Jakub Łuszczek & PrimeGrid | 08 May 2011 |
126667 | 626497 | Ray Ballinger | 09 Jun 2003 |
130139 | 280296 | Dale Andrews | 02 Feb 2002 |
141941 | 4299438 | Scott Brown & PrimeGrid | 26 May 2011 |
144643 | 498079 | Richard Heylen | 12 Dec 2000 |
148901 | 360338 | Mark Rodenkirch | 05 Mar 2002 |
149797 | 1414137 | Peter van Hoof & RSP | 13 Mar 2005 |
150847 | 1076441 | Darren Wallace & RSP | 15 Aug 2004 |
152713 | 1154707 | Ray Ballinger | 23 Oct 2004 |
159371 | 284166 | Janusz Szmidt | 14 Jan 2002 |
170591 | 866870 | Drew Bishop & RSP | 15 Apr 2004 |
189463 | 324103 | Dave Linton | 15 Jul 2000 |
191249 | 3417696 | Jonathan Pritchard & PrimeGrid | 21 Nov 2010 |
192089 | 1395688 | Guido Stolz & RSP | 10 May 2004 |
196597 | 2178109 | Auritania Du & RSP | 09 May 2006 |
201193 | 457615 | Daval Davis | 03 Feb 2003 |
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 |
220063 | 306335 | Olivier Haeberlé | 03 Sep 1999 |
222997 | 613153 | Olivier Haeberlé | 28 Nov 2001 |
234847 | 1535589 | Darren Wallace & RSP | 09 May 2005 |
235601 | 295338 | Helmut Zeisel | 06 Mar 2003 |
245051 | 285750 | Tom Kuechler | 15 Nov 2000 |
246299 | 752600 | Kevin O'Hare & RSP | 23 Jan 2004 |
252191 | 5497878 | Jan Haller & PrimeGrid | 23 Jun 2012 |
261221 | 689422 | Sean Faith & RSP | 22 Dec 2003 |
267763 | 264115 | Dave Linton | 19 Feb 2000 |
275293 | 2335007 | Japke Rosink & RSP | 21 Sep 2006 |
277153 | 429819 | Jeff Wolfe | 21 Nov 2002 |
279703 | 616235 | Dhumil Zaveri & RSP | 07 Jan 2004 |
299617 | 428917 | Dave Linton | 22 Jul 2002 |
304207 | 6643565 | Randy Ready & PrimeGrid | 11 Oct 2013 |
309817 | 901173 | Helmut Michel & RSP | 07 Jun 2004 |
325627 | 1472117 | Will Fisher & RSP | 05 Apr 2005 |
342673 | 2639439 | Dhumil Zaveri & RSP | 28 Apr 2007 |
345067 | 1876573 | Dave Linton | 13 Nov 2005 |
350107 | 1144101 | Sean Faith & RSP | 24 Oct 2004 |
353159 | 4331116 | Jaakko Reinman & PrimeGrid | 31 May 2011 |
357491 | 609338 | Lucas Schmid | 17 Jan 2003 |
357659 | 1779748 | Drew Bishop & RSP | 25 Sep 2005 |
376993 | 293603 | Reto Keiser | 08 Sep 2002 |
382691 | 431722 | Ray Ballinger | 27 Feb 2003 |
398023 | 6418059 | Vladimir Volynsky & PrimeGrid | 05 Oct 2013 |
398533 | 419107 | Dave Linton | 04 Sep 2002 |
401143 | 532927 | Olivier Haeberlé | 11 Jun 2003 |
401617 | 470149 | Dave Linton | 27 Dec 2002 |
412717 | 1084409 | Holger Meissner & RSP | 22 Aug 2004 |
415267 | 3771929 | Alexey Tarasov & PrimeGrid | 08 May 2011 |
416413 | 424791 | Dave Linton | 28 Apr 2003 |
417643 | 1800787 | Greg Childers & RSP | 05 Oct 2004 |
428639 | 3506452 | Brett Melvold & PrimeGrid | 14 Jan 2011 |
443857 | 369457 | Nuutti Kuosa | 27 Aug 2001 |
450457 | 2307905 | Jeff Smith & RSP | 28 Mar 2006 |
458743 | 547791 | Olivier Haeberlé | 22 Oct 2003 |
460139 | 779536 | Drew Bishop & RSP | 26 Mar 2004 |
465869 | 497596 | Lucas Schmid | 27 Jan 2003 |
467917 | 1993429 | Steven Wong & RSP | 25 Dec 2005 |
469949 | 1649228 | Steven Wong & RSP | 28 Oct 2007 |
485767 | 3609357 | Chris Cardall & RSP | 24 Jun 2008 |
500621 | 1138518 | Darren Wallace & RSP | 18 Oct 2004 |
502541 | 1199930 | Ryan Sefko & RSP | 21 Dec 2004 |
504613 | 1136459 | Magnus Mischel & RSP | 17 Oct 2004 |
REFERENCES:
Ballinger, R. and Keller, W. "The Riesel Problem: Search for Remaining Candidates." https://www.prothsearch.net/rieselsearch.html.
Keller, W. "The Riesel Problem: Definition and Status." https://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." https://www.rieselsieve.com/.
Sloane, N. J. A. Sequences A046067, A046068, A046069, and A046070 in "The On-Line Encyclopedia of Integer Sequences."
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