Riesel Number

There exist infinitely many odd integers k such that k·2^n-1 is composite for every n>=1. Numbers k 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 k=509203, but as of Jan. 2014, there remain 52 smaller candidates which generate only composite numbers for all n 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 k=509203 is the smallest Riesel number is sometimes known as the Riesel problem or Riesel conjecture.

Let a(k) be smallest n for which (2k-1)·2^n-1 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 n 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 k·2^n-1 discovered to date providing disproof of the existence of smaller Riesel numbers are summarized in the following table (Keller).

See also

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References

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Cite this as:

Weisstein, Eric W. "Riesel Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RieselNumber.html

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