A solution to the Straus-Erdős conjecture

TL;DR

The paper addresses the Straus–Erdős conjecture, which posits that for every prime $p$ there exist positive integers $x \le y \le z$ with $4/p = 1/x + 1/y + 1/z$. It develops an elementary, constructive framework by partitioning potential solutions into Type I (p does not divide y) and Type II (p divides y) and deriving a parametric structure: for $p ≡ 1 mod 4$, Type I forces $z = ((4k+3)p^2 + p)/4$ and Type II yields $x = (p + (4k+3))/4$, with both reduced to two-term unit-fraction decompositions driven by $k$ and an auxiliary parameter. By selecting $k$ and an auxiliary $ ext{ell}$ and enforcing modular constraints, the method builds explicit unit-fraction identities and assembles them into a covering system across primes described by several residue classes (e.g., $p ≡ 29 mod 44$, $p ≡ 41 mod 44$, etc.). The section provides concrete $k=0$ instances to illustrate these decompositions and demonstrates how the covering-system framework could yield a complete constructive proof. If validated, this approach offers a novel elementary route to resolving the conjecture via modular prime coverings and explicit fraction identities.

Abstract

This paper outlines a solution to the Straus Erdős Conjecture. Namely for each prime $p$ there exists positive integers $x \leq y \leq z$ so that $$ \frac{4}{p} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z} $$

Theorems & Definitions (5)

• Proposition 1
• proof
• Lemma 1
• Proposition 2
• Lemma 2