Infinitely Many Ramanujan-Type Congruences for p_{3,1}(n) Modulo 3 - Elementary Reduction and Conjectural Framework

Abstract: Let p_{3,1}(n) denote the number of partitions of n into distinct parts with a parity or modulo condition related to G(q) = (-q;q^2)_inf / (q^2;q^2)_inf. I have proved a factorization G(q) = H(q)A(q^3) where H(q) and A(q) are explicit q-series. I proved the base case p_{3,1}(6) ≡ 0 (mod 3) by elementary coefficient extraction. Using the factorization, I proposed the conjectural infinite family p_{3,1}(3^k n + δ_k) ≡ 0 (mod 3) for k ≥ 1, with explicit δ_k. Full proof of the infinite family requires further analysis of the 3-adic properties of A(q). This work establishes the elementary reduction and provides a framework for future work on Ramanujan-type congruences. Keywords: Integer partitions, q-series, Ramanujan congruences, p_{3,1}(n), q-Pochhammer symbol Author: Sayan Kumar Mitra, Class 11, Rajendra Vidyalaya, Jamshedpur, India