About the Riesel Problem

Hans Ivar Riesel (May 28, 1929 in Stockholm - December 21, 2014) was a Swedish mathematician. In 1956, he showed that there are an infinite number of positive odd integer k's such that k*2^n-1 is composite (not prime) for every integer n>=1. These numbers are now called Riesel numbers. He further showed that k=509203 was such one.

It is conjectured that 509203 is the smallest Riesel number. The Riesel problem consists of determining that 509203 is the smallest Riesel number. To show that it is the smallest, a prime of the form k*2^n-1 must be found for each of the positive integer k's less than 509203. As of November 17th, 2020, there remain 48 k's for which no primes have been found. They are as follows:

2293, 9221, 23669, 31859, 38473, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 161669, 192971, 206039, 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

For a more detailed history and status of the Riesel problem, please visit Wilfrid Keller's The Riesel Problem: Definition and Status.

To participate in effort, please select The Riesel Problem (LLR) project in your PrimeGrid preferences section.

Additional Information

• Riesel Number (Wolfram MathWorld)
  
• Riesel Number (Wiki)
  
• Riesel Number (The Prime Glossary)
  

The Riesel problem is to k*2^n-1 as the Sierpinski problem is to k*2^n+1. There is no equivalent to the 'prime' Sierpinski problem since k=509203, the conjectured smallest Riesel number, is prime.

Primes found at PrimeGrid

191249*2^3417696-1 by Jonathan Pritchard on 21 Nov 2010. Official Announcement
428639*2^3506452-1 by Brett Melvold on 14 Jan 2011. Official Announcement
65531*2^3629342-1 by Adrian Schori on 5 Apr 2011. Official Announcement
123547*2^3804809-1 by Jakub Łuszczek on 8 May 2011. Official Announcement
415267*2^3771929-1 by Alexey Tarasov on 8 May 2011. Official Announcement
141941*2^4299438-1 by Scott Brown on 26 May 2011. Official Announcement
353159*2^4331116-1 by Jaakko Reinman on 31 May 2011. Official Announcement
162941*2^993718-1 by Dmitry Domanov on 02 February 2012. Official Announcement
252191*2^5497878-1 by Jan Haller on 23 June 2012. Official Announcement.
398023*2^6418059-1 by Vladimir Volynsky on 5 October 2013. Official Announcement.
304207*2^6643565-1 by Randy Ready on 10 October 2013. Official Announcement.
40597*2^6808509-1 by Frank Meador on 25 December 2013. Official Announcement.
402539*2^7173024-1 by Walter Darimont on 2 October 2014. Official Announcement.
502573*2^7181987-1 by Denis Iakovlev on 4 October 2014. Official Announcement.
273809*2^8932416-1 by Wolfgang Schwieger on 13 December 2017. Official Announcement.
146561*2^11280802-1 by Pavel Atnashev on 17 November 2020. Official Announcement.
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