The Riesel Problem: Definition and Status

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.

To prove the conjecture, it suffices to exhibit a prime k · 2n − 1 for each k < 509203. This has yet to be accomplished for the following 43 values of k:

    23669,   31859,   38473,   46663,   67117,   74699,   81041,   97139, 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 December 09, 2022.

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.

In order to summarize the known results, let us define fm to be the number of multipliers k < 509203 giving their first prime k · 2n − 1 for an exponent n in the interval 2m < n < 2m+1. Then f0 = 39867 is the number of those k for which k · 2 − 1 is a prime, the first one being k = 3 (note that 1 · 2 − 1 = 1 is not considered to be a prime). More generally, the following frequencies have been determined so far:

  m fm
0   39867
1 59460
2 62311
3 45177
4 24478
5 11668
6 5360
7 2728
8 1337
9 785
10 467
11   289
         
  m fm
12   191
13 125
14 87
15 62
16 38
17 35
18 25
19 22
20 18
21 13
22 8
23 ≥ 7

References.

  • H. Riesel, Några stora primtal (Swedish: Some large primes), Elementa 39 (1956), 258-260.
  • W. Keller, Factors of Fermat numbers and large primes of the form k · 2n + 1, II, unpublished manuscript, Hamburg, September 1992.
  • Y. Gallot, unpublished data, 1998.
  • Y. Gallot, On the number of primes in a sequence, http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf, 2001.

    For more information see the Riesel number page in Chris Caldwell's Glossary.

    Address questions about this web page to Wilfrid Keller

    URL: http://www.prothsearch.com/rieselprob.html
    Last modified: December 09, 2022.