Riesel number - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an A composite number is a positive Integer which has a positive Divisor other than one or itself

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. Year 1956 ( MCMLVI) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. Hans Ivar Riesel (born 1929 in Stockholm) is a Swedish Mathematician who discovered the 18th known Mersenne prime in 1957 Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

A number can be shown to be a Riesel number by giving its "covering set". In Mathematics, a covering set for a Sequence of Integers refers to a set of Prime numbers such that every term in the sequence 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). Riesel Sieve is a Distributed computing project running in part on the BOINC platform Seventeen or Bust is a Distributed computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem. In Number theory, a Sierpinski number is an odd Natural number k such that integers of the form k 2 n + 1 are composite

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This number theory-related article is a stub. In Number theory, a Sierpinski number is an odd Natural number k such that integers of the form k 2 n + 1 are composite Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes You can help Wikipedia by expanding it.

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