A theorem on partitions

Abstract

Certain integers have the property that they can be partitioned into distinct positive integers whose reciprocals sum to 1, e.g., and In this paper we prove that all integers exceeding 77 possess this property. This result can then be used to establish the more general theorem that for any positive rational numbers α and β there exists an integer r(α, β) such that any integer exceeding r(α, β) can be partitioned into distinct positive integers exceeding β whose reciprocals sum to α.