Infinitely Many Ramanujan-type Congruences for p_{4,1}(n) mod 7 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur
Authors/Creators
- 1. Rajendra Vidyalaya, Jamshedpur, Jharkhand, India - Class 11 Student (Independent Researcher)
Description
UPDATE - v2 is corrected version.
This v2 corrects all typos from v1. $p_{4,1}(4) = 63$ (not 7), other values corrected to 4928, 129976. Generating-function decomposition on Pages 1-2 is fully corrected in this version. The divisibility by 7 holds for all corrected values.
Author: Sayan Kumar Mitra, Class 11, Rajendra Vidyalaya, Jamshedpur, Jharkhand, India.
Independent Researcher in Number Theory.
TITLE: Infinitely Many Ramanujan-type Congruences for p_{4,1}(n) modulo 7
ABSTRACT:
Let p_{4,1}(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 p_{4,1}(n) ≡ 0 mod 7.
The proof uses the theory of modular forms modulo 7. I show that the generating function G(q) for p_{4,1}(n) is congruent modulo 7 to a product H(q) = ∏_{m≥1} 1/(1 - q^{m}) 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 p_{4,1}(l^k n + δ_k) ≡ 0 mod 7 for primes l.
Keywords: p41 partitions, partitions mod 7, Ramanujan congruences, Sayan Kumar Mitra, Rajendra Vidyalaya Jamshedpur, Number Theory
High School Research Paper