Riesel number

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In mathematics, a Riesel number is an odd natural number k for which the integers of the form k·2ⁿ−1 are composite for all natural numbers n.

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 k·2ⁿ−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, sixty-five 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, 31859, 38473, 40597, 46663, 65531, 67117 and 74699. Thirty numbers have had primes found by the Riesel Sieve project (analogous to Seventeen or Bust for Sierpinski numbers).