Rogers-Ramanujan Identities Link Partitions to Eta Products via Mod 5 Congruence — E8 Intelligence Research

FINDING: The Rogers-Ramanujan identities connect integer partitions with distinct parts to partitions with parts differing by ≥2, via a product-sum identity; the mod 5 congruence is a direct consequence of the Dedekind eta product structure. | MATH: - **Rogers-Ramanujan identities** (analytic form): \\[ \\sum_{n=0}^\\infty \\frac{q^{n^2}}{(q;q)_n} = \\frac{1}{(q;q^5)_\\infty (q^4;q^5)_\\infty}, \\quad \\sum_{n=0}^\\infty \\frac{q^{n^2+n}}{(q;q)_n} = \\frac{1}{(q^2;q^5)_\\infty (q^3;q^5)_\\infty} \\] where \\((a;q)_n = \\prod_{k=0}^{n-1}(1-aq^k)\\). - **Combinatorial form**: partitions into parts differing by ≥2 (first) or ≥2 with no 1's (second) equal partitions into parts ≡ ±1 (mod 5) or ±2 (mod 5). - **Mod 5 congruence** (Ramanujan): \\(p(5n+4) \\equiv 0 \\pmod{5}\\). This follows from the eta product: \\[ \\sum_{n=0}^\\infty p(5n+4) q^n = 5 \\frac{(q^5;q^5)_\\infty^5}{(q;q)_\\infty^6} \\] which is a modular form of weight 2 on \\(\\Gamma_0(5)\\). - **Dedekind eta**: \\(\\eta(q) = q^ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com