The Full Divisor Criterion for the Erdős–Straus Conjecture: Covering-System Closures and Their Limits

The Erdős–Straus conjecture asserts that for every integer 𝑛 ≥ 2 the Diophantine equation4/n=1/x+1/y+1/z admits a solution in positive integers 𝑥, 𝑦, 𝑧. Following the classical reductionsof Mordell, the conjecture is known to hold for every 𝑛 outside a thin system of residue classesmodulo 840, generated by the six residues 𝑛 ≡ 1, 121, 169, 289, 361, 529 (mod 840) — each aperfect square, and each therefore immune to the polynomial congruence identities that resolveevery other class. In this paper we develop two complementary tools aimed at this residual class.First, we establish the Full Divisor Criterion (Theorem 7), a necessary-and-sufficient condition,in terms of a divisor search inside 𝑥2, for a prime 𝑛 in one of these six classes to be good. Second,building on this criterion, we derive several closed-form families (Theorems 8–10) and combinethem with an extensive independent covering-system computation, across the primes 5, 7, 11, 13,17, 19, and 23, which together close more than 98.5% of the residue classes making up this finalcase, together with a genuine, unconditional partial theorem (the 𝐷 = 1 criterion) that resolvesa substantial share of cases outright with no search required. Three small combined systemsare enumerated completely: mod (5×7×11) leaves 36 exceptional classes, mod (5×7×13) leaves52, and mod (5×7×23) leaves 78. We additionally report a mechanical search that appliesthe Full Divisor Criterion directly, well beyond the range of the hand-built families above; itreproduces every documented closure but finds no new ones, a negative result reported honestlyrather than omitted. What remains is a residual set of density under 1.5%, for which directcomputation finds a solution in every case tested but for which no unconditional covering oranalytic argument is yet supplied. This paper therefore falls short of a complete proof of theErdős–Straus conjecture; what it offers instead is a fully explicit reduction of the conjectureto this residual set, a verified toolkit for attacking it, and, in the closing sections, an honestaccount of the gap that remains.