On Representations of Reciprocals as a Sum of Three Distinct Unit Fractions

Abstract: We study representations of reciprocals of positive integers in the form $$1/n = 1/a + 1/b + 1/c$$, where $$n < a < b < c$$. We derive explicit bounds for the variables and present two transformations reducing the problem to a two-variable setting and to a divisor-based parametrization. While the latter provides structural insight via the factorization of $$m^2$$, it does not capture all admissible solutions. Therefore, we propose a direct enumeration algorithm that examines $$O(n^2 \log n)$$ candidate pairs for each fixed $$n$$ and guarantees complete results. Numerical computations for $$1 \leq n \leq 100$$ illustrate the irregular behavior of the number of representations and its dependence on the arithmetic structure of $$n$$.