Erdős–Straus Conjecture: Modular Constraints and the Remaining Prime Case — E8 Intelligence Research

FINDING: The Erdős–Straus conjecture (4/n = 1/x + 1/y + 1/z for all n≥2) remains unsolved; known modular constraints reduce the problem to prime n ≡ 1 mod 24, with partial residue-class proofs but no complete classification. | MATH: Core equation: 4/n = 1/x + 1/y + 1/z, x,y,z ∈ ℕ. For prime p, the standard reduction: if p ≡ 2 mod 3, solution exists via (p+1)/2, p(p+1)/2, p(p+1)/2; if p ≡ 3 mod 4, via (p+1)/4, p(p+1)/4, p(p+1)/2. Remaining hard case: p ≡ 1 mod 24 (since p ≡ 1 mod 4 and p ≡ 1 mod 3). Known parametric families: e.g., for p = 24k+1, one can use x = 6k+1, y = p(6k+1), z = p(6k+1)/(4k) — but only when 4k divides p(6k+1), which is not always. The arxiv note (1906.00561) claims a fundamental structural assertion under x≤y≤z, but its validity is contested. | CONNECTION: The modular structure (mod 24, mod 3, mod 4) echoes base-60/12-fold symmetries — the number 24 is the order of the binary tetrahedral group and the kissing number in 4D; the residue classes form a lattice in ℤ/2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com