Sierpinski and Riesel Numbers
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) 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 the covering set.
These covering sets form from prime numbers that in base 2 have short periods. To achieve a complete covering set, it can be shown that the sequence can repeat no more frequently than every 24 numbers. A repeat every 24 numbers give the covering set {3, 5, 7, 13, 17, 241} , while a repeat every 36 terms can give several covering sets: {3, 5, 7, 13, 19, 37, 73}; {3, 5, 7, 13, 19, 37, 109}; {3, 5, 7, 13, 19, 73, 109} and {3, 5, 7, 13, 37, 73, 109}. Riesel numbers have the same covering sets as Sierpinski numbers.
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“What culture lacks is the taste for anonymous, innumerable germination. Culture is smitten with counting and measuring; it feels out of place and uncomfortable with the innumerable; its efforts tend, on the contrary, to limit the numbers in all domains; it tries to count on its fingers.”
—Jean Dubuffet (1901–1985)