# Wild and Wooley Numbers

by Jeffrey C. Lagarias

Year of Award: 2007

Award: Lester R. Ford

Publication Information: The American Mathematical Monthly, vol. 113, (2006), pp. 97-108

Summary: (from the author's abstract)  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.