Infinitely Many Ramanujan-type Congruences for p_{4,1}(n) mod 7 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur
- 1Rajendra Vidyalaya, Jamshedpur, Jharkhand, India - Class 11 Student (Independent Researcher)
Abstract
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
Related
- Ramanujan-type congruences for the smallest parts function
- Generalized Ramanujan Congruences for Arbitrary-Weight Modular Forms — E8 Intelligence Research
- Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta-functions
- Identities and congruences involving orthogonal polynomials and Apéry-like numbers
- Degeneration of monodromy in the theory of Eisenstein congruences for classical groups
BibTeX
@misc{mitra2026infinitely,
title = {{Infinitely Many Ramanujan-type Congruences for p_4,1(n) mod 7 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur}},
author = {Sayan Kumar Mitra},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23465776},
url = {https://zenodo.org/records/23465776},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinitely Many Ramanujan-type Congruences for p_{4,1}(n) mod 7 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur
AU - Sayan Kumar Mitra
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23465776
UR - https://zenodo.org/records/23465776
AB - 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
ER -