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 is divisible by at least one member of the set. The term "covering set" is used only in conjunction with sequences possessing exponential growth.
The use of the term "covering set" is related to Sierpinski and Riesel numbers. These are odd natural numbers k for which the formula k 2n + 1 (Sierpinski number) or k 2n − 1 (Riesel number) produces no prime numbers. Since 1960 it has been known that there exists an infinite number of both Sierpinski and Riesel numbers (as solutions to families of congruences based upon the set {3, 5, 17, 257, 641, 65537, 6700417} ) but, because there are an infinitude of numbers of the form k 2n + 1 or k 2n − 1 for any k, one can only prove k to be a Sierpinski or Riesel number through showing that every term in the sequence k 2n + 1 or k 2n − 1 is divisible by one of the prime numbers of a covering set.
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