Riesel number

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In mathematics, a Riesel number is an odd natural number k for which the integers of the form k2ⁿ-1 are all composite.

In other words, when k is a Riesel number, all members of the following set are composite:

$\left\{\,k 2^n - 1 : n \in\mathbb{N}\,\right\}$

In 1956, Hans Riesel showed that there are an infinite number of integers k such that k2ⁿ-1 is not prime for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810.

A number can be shown to be a Riesel number by giving its "covering set". A covering set is a set of small prime numbers that will divide any member of a sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have the following covering sets:

509203×2ⁿ-1 has covering set {3, 5, 7, 13, 17, 241}
762701×2ⁿ-1 has covering set {3, 5, 7, 13, 17, 241}
777149×2ⁿ-1 has covering set {3, 5, 7, 13, 19, 37, 73}
790841×2ⁿ-1 has covering set {3, 5, 7, 13, 19, 37, 73}
992077×2ⁿ-1 has covering set {3, 5, 7, 13, 17, 241}

The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured that 509203 is the smallest Riesel number. However, 69 values of k less than this have yielded only composite numbers for all values of n so far tested. The smallest of these are 2293, 9221, 23669, 26773, 31859, 38473, 40597, 46663, 65531, 67117 and 74699. Twenty-six numbers have had primes found by the Riesel Sieve project (analogous to Seventeen or Bust for Sierpinski numbers).