On Egyptian fractions of length 3

Abstract

Let a, n be positive integers that are relatively prime. We say that a/n can be represented as an Egyptian fraction of length k if there exist positive integers m1, . . . ,mk such that a n = 1 m1 + · · ·+ 1 mk . Let Ak(n) be the number of solutions a to this equation. In this article, we give a formula for A2(p) and a parametrization for Egyptian fractions of length 3, which allows us to give bounds to A3(n), to fa(n) = #{(m1,m2,m3) : a n = 1 m1 + 1 m2 + 1 m3 }, and finally to F (n) = #{(a,m1,m2,m3) : a n = 1 m1 + 1 m2 + 1 m3 }.