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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.〔Guy, Richard; ''Unsolved Problems in Number Theory''; pp. 119–121. ISBN 0387208607〕 The term "covering set" is used only in conjunction with sequences possessing exponential growth.
==Sierpinski and Riesel numbers==
The use of the term "covering set" is related to Sierpinski and Riesel numbers. These are odd natural numbers for which the formula (Sierpinski number) or (Riesel number) produces no prime numbers.〔Wells, David; ''Prime Numbers: The Most Mysterious Figures in Math''; pp. 212, 219. ISBN 1118045718〕 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 〔Sierpiński, Wacław (1960); ‘Sur un problème concernant les nombres’; ''Elemente der Mathematik'', 15(1960); pp. 73–96〕) but, because there are an infinitude of numbers of the form or for any , one can only prove to be a Sierpinski or Riesel number through showing that ''every'' term in the sequence or is divisible by one of the prime numbers of a covering set.
These covering sets form from prime numbers that in base 2 have short periods. To achieve a complete covering set, Wacław Sierpiński showed that a sequence can repeat no more frequently than every 24 numbers. A repeat every 24 numbers give the covering set , while a repeat every 36 terms can give several covering sets: ; ; and .〔(Covering Sets for Sierpiński Numbers )〕
Riesel numbers have the same covering sets as Sierpinski numbers.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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