Mathematics in Education, Research and Applications (MERAA), 2025(11), 1
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Published online 2025-12-10
DOI:https://doi.org/10.15414/meraa.2025.11.01.9-17
Decompositions of 1⁄n into sums and differences of reciprocals of the forms 2⁄a and 3⁄b
Radovan Potůček
University of Defence, Brno, Czech Republic
Article Fulltext (PDF), pp. 9–17
- This paper addresses the elementary problem of decomposing the reciprocal of a positive integer into a specific sum or difference of two reciprocals
and computing their total number. We show that, for any positive integer n, the number of decompositions of 1⁄n into the form 2⁄a ± 3⁄b is determined by the number
of positive divisors of 6n², and we derive general results together with explicit formulas. The method is illustrated by explicit decompositions for a chosen positive
integer. Finally, we describe a program implemented in the computer algebra system Maple 2025 that computes these decompositions for any positive integer, and we confirm the theoretical results using the earlier example.
- Keywords: reciprocal, Egyptian fraction, greedy algorithm, Erdős–Straus conjecture, prime factorization, Simon’s favorite factoring trick, Diophantine equation, divisor function, computer algebra system Maple
- JEL Classification: C60