Counting the number of solutions to the erdås-straus equation on unit fractions

Abstract

For any positive integer n, let f(n) denote the number of solutions to the Diophantine equation with x, y, z positive integers. The ErdÅ's-Straus conjecture asserts that f(n)> 0 for every n≥ 2. In this paper we obtain a number of upper and lower bounds for f(n) or f(p) for typical values of natural numbers n and primes p. For instance, we establish that hspace0.167em 2 N ∑ Nf(p) N 0.167em 2 N N. These upper and lower bounds show that a typical prime has a small number of solutions to the ErdÅ's-Straus Diophantine equation; small, when compared with other additive problems, like Waring's problem. © 2013 Australian Mathematical Publishing Association Inc.