Infinitely Many Ramanujan-type Congruences for p4,1(n) mod 7 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur

Author: Sayan Kumar Mitra, Class 11, Rajendra Vidyalaya, Jamshedpur, Jharkhand, India. Independent Researcher in Number Theory. TITLE: Infinitely Many Ramanujan-type Congruences for p4,1(n) modulo 7 ABSTRACT: Let p4,1(n) denote the number of partitions of n where parts ≡ ±1 mod 4 are distinct and counted with specific weights. In this paper, I proved that there are infinitely many n such that p4,1(n) ≡ 0 mod 7. The proof uses the theory of modular forms modulo 7. I show that the generating function G(q) for p4,1(n) is congruent modulo 7 to a product H(q) = ∏m≥1 1/(1 - qᵐ) with modifications, which exhibits modular properties. This is related to Ramanujan's famous partition congruences p(5n+4) ≡ 0 mod 5. Conjecture: There exists an infinite family of congruences of the form p4,1(l^k n + δ_k) ≡ 0 mod 7 for primes l.