What question did this study set out to answer?

To develop a deterministic algorithm that produces an Egyptian fraction expansion for 4/n without external references to existing conjectures.

February 6, 2026Open Access

An Erdős-Straus divisor–lattice construction for 4/n

Key Points

To develop a deterministic algorithm that produces an Egyptian fraction expansion for 4/n without external references to existing conjectures.
Utilized a greedy algorithm on the divisor lattice of the least common multiple (LCM) of integers.
Applied elementary divisor arithmetic in conjunction with the harmonic sum.
Generated a canonical triple (Xn, Yn, Zn) for each integer n ≥ 2.
Successfully produced Egyptian fraction expansions for all integers n ≥ 2.
Established a natural counting method for representations derived from the divisor lattice.
Avoided reliance on the Erdős–Straus conjecture, providing a novel approach.

Abstract

We present a deterministic greedy algorithm on the divisor lattice of Ln = lcm (1, 2, . . . , n)that, for every integer n ≥ 2, produces an Egyptian fraction expansion 4/n = 1/Xn+1/Yn+1/Zn ; with Xn, Yn, Zn ∈ Z>0. The construction uses only elementary divisor arithmetic and the harmonic sum Hn = ∑ 1/k, 1≤k≤n without invoking any external results on the Erdős–Straus conjecture. The method yields an explicit, canonical triple (Xn, Yn, Zn) for each n, and provides a natural counting of the number of such representations obtainable from the divisor lattice.

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