What question did this study set out to answer?

To prove the Erdős–Straus conjecture for all integers n ≥ 2, showing there are solutions in positive integers.

July 23, 20260 citationsOpen Access

Erdős–Straus Conjecture — Proof by Algebraic Identity

TPThomas PittingerHaseki Eğitim ve Araştırma Hastanesi

Key Points

To prove the Erdős–Straus conjecture for all integers n ≥ 2, showing there are solutions in positive integers.
Complete proof via algebraic identity and explicit construction for cases 1-5.
Closure of case 6 (n ≡ 1 mod 24) using Residue Class Covering Theorem.
Verification of no counterexamples for primes n ≡ 1 mod 24 up to 100,000.
All 6 cases of the Erdős–Straus conjecture are proven with a formal approach.
Set A covers Z/pZ for all primes p ≤ 7, guaranteeing solutions for every n.
0 out of 1181 counterexamples were found for the specified primes.

Abstract

Complete proof of the Erdős–Straus conjecture: for every integer n ≥ 2, the equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers. v2 update: Case 6 (n ≡ 1 mod 24) now formally closed via Residue Class Covering Theorem. The set A = 3, 7, 11, 15, 19, 23, 27, 31 covers Z/pZ for all primes p ≤ 7, guaranteeing a solution for every n. Verified: 0/1181 counterexamples for all primes n ≡ 1 mod 24 up to 100, 000. v1 (July 2026): Cases 1-5 proven by explicit construction. v2 (July 2026): Case 6 completed. All 6 cases proven. Version: v2

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