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 (sequence A101036 in OEIS).

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.[1]

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

  • 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, 52 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, 46663, 67117, 74699, 81041 and 93839.

Thirty-three numbers have had primes found by the Riesel Sieve project (analogous to Seventeen or Bust for Sierpinski numbers). Currently, PrimeGrid is working on the remaining numbers and has found twelve primes as of 25 December 2013.[2]

The smallest n for which k*2^n-1 is prime.[edit]

(sequence A040081 in OEIS)

k 1 2 3 4 5 6 7 8 9 10 11 12 13 ... 59 ... 127 ... 148 ... 191 ... 659 ... 2293 ... 9221 ... 23669 ... 40597 ... 509203 ...
smallest n 2 1 0 0 2 0 1 0 1 1 2 0 3 ... 12 ... 25 ... 2551 ... 226 ... 800516 ... ? ... ? ... ? ... 6808509 ... none ...

Simultaneously Riesel and Sierpiński[edit]

A number may be simultaneously Riesel and Sierpiński. These are called Brier numbers. The smallest known example (as of March 2013) is 143665583045350793098657 (A076335).[3]

See also[edit]

References[edit]

  1. ^ Riesel, Hans (1956). "Några stora primtal". Elementa 39: 258–260. 
  2. ^ Twelfth Riesel prime discovery announcement on PrimeGrid
  3. ^ Problem 29.- Brier Numbers

Sources[edit]

External links[edit]