The Riesel Problem: Definition and Status

The Riesel Problem: Definition and Status

In 1956 Hans Riesel 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 Sierpinski numbers. 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. A reasonable approach to the 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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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 kfor 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:

  m fm
0   39867
1 59460
2 62311
3 45177
4 24478
5 11668
6 5360
7 2728
8 1337
9 785
  m fm
10   467
11 289
12 191
13 125
14 87
15 62
16 38
17 35
18 25
19 21
  • The frequencies fm for m < 10 were first computed by Wilfrid Keller in 1992 and independently verified by Yves Gallot, who extended the computations to m < 13. This range was finished on May 11, 1998.
  • Later on, Ray Ballinger, Wilfrid Keller, and Skip Key determined that f14 = 87, by finding 21, 25, and 41 first primes, respectively, having 16384 < n < 32768. These computations were finished on November 14, 1998.
  • On April 9, 1999, the value of f15 = 62 (corrected May 12, 1999, and April 20, 2000) was established. The corresponding primes k.2n – 1 with 32768 < n < 65536 were found by Ray Ballinger (5 primes), Kenneth Brazier (1 prime), Chris Caldwell (2 primes), Wilfrid Keller (13 primes), Skip Key (22 primes), Tom Kuechler (1 prime), and Dave Linton (18 primes). Other contributors to this segment were David Anderson, Chad Davis, Yves Gallot, Michal Misztal, Anton Oleynick, Michael Peake, Janusz Szmidt, and Helmut Zeisel.  Home page