The monoid of numbers of the form 1 < a^q /b^p < a

Abstract

This paper is a study of the set of rational numbers of the form 1 < a^q /b^p < a with a and b co-prime integers. The set F (a,b) of these numbers, with an appropriate binary law, is a monoid isomorphic to (N, +, 0). We identify the sequences of minimum and maximum record holders in F (a,b) and prove that the first one converges to 1 while the second one converges to a. We conclude that F (a,b) is dense in the set of the real numbers comprise between 1 and a.