On the Decompositions of a Reciprocal Into the Sum and the Difference of Two Reciprocals

Abstract: This paper deals with the relatively simple problem of determining the decomposition of a reciprocal into the sum or difference of two reciprocals, and of establishing how many such decompositions exist. First, we demonstrate the method on four illustrative cases. We then show that, for a positive integer n, the number of decompositions of 1/n into the sum or difference of two reciprocals is determined by the number of divisors of n2. Afterwards, we illustrate the process of finding decompositions for a specific positive integer. Next, we present the results for an arbitrary positive integer and derive the corresponding explicit formulas. Finally, we present a program created in the computer algebra system Maple 2025 for determining the decompositions for any positive integer, and verify the theoretical results for the previously considered example.