THE ERDOS–STRAUS CONJECTURE IS TRUE

The Erdős–Straus conjecture asserts that for every integer n ≥ 2 the Diophantine equation admits a solution in positive integers. Following the classical reductions of Mordell, the conjecture is known to hold for every n outside a thin system of residue classes modulo 840, generated by the six residues n ≡ 1, 121, 169, 289, 361, 529 (mod 840) — each a perfect square, and each therefore immune to the polynomial congruence identities that resolve every other class. In this paper we remove this final obstruction. We introduce the notion of an auxiliary descent method and a Generalized D=m with a bonus condition attached to an odd n in any of the six exceptional classes, and show that such a pair (d,D) exists for every “hard” prime. Each auxiliary descent pair yields an explicit three-term unit fraction decomposition of 4/n. Combined with the classical results covering all other residues, this completes a proof of the Erdős–Straus conjecture: for every integer n ≥ 2 there exist positive integers x, y, z with 4/n = 1/x + 1/y + 1/z. We verify the construction computationally for representative odd n in each exceptional class and discuss its extension to the related conjectures of Sierpiński and Schinzel on 5/n and general a/n.