## 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 44 values of k:

 23669,   31859,   38473,   46663,   67117,   74699,   81041,   93839,   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 April 27, 2021.

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 ≥ 6

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• W. Keller, Factors of Fermat numbers and large primes of the form k · 2n + 1, II, unpublished manuscript, Hamburg, September 1992.
• Y. Gallot, unpublished data, 1998.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.