Wild and Wooley Numbers

Details Wild and Wooley Numbers Wild and Wooley Numbers

2004-11-07

arxiv.org/abs/math/0411141v2

American Mathematical Monthly 113 (2006), 97--108.

Mathematics

Number Theory

11 pages, latex; revised with title change and some notation changes

Scientific paper

This paper studies the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n, together with 1/2. The subsemigroup of integers in this semigroup is called the wild integer semigroup, and the wild numbers are the irreducible elements in this subsemigroup. This paper presents evidence that the wild numbers consist of all the prime numbers p except 3. The subsemigroup of integers in the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n is called the Wooley integer semigroup and its irreducible elements are called Wooley numbers. This semigroup is shown to be recursive, and various open problems are formulated about Wooley integers.

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