Congruences via Partitions with Exactly Two Part Sizes

TL;DR

This paper proves a modular congruence involving divisor sums and partitions with two part sizes for specific linear forms of N, using combinatorial and number-theoretic methods.

Contribution

It establishes a new congruence relation for divisor sums linked to partitions with exactly two part sizes, extending previous combinatorial and modular arithmetic results.

Findings

Proves a congruence involving divisor sums for specific linear forms of N.

Connects partition counts with exactly two part sizes to divisibility properties.

Uses a combination of combinatorial and modular techniques to establish the result.

Abstract

We prove the congruence $\sum_{1 \leq k < \sqrt{N}} \sigma_0 (N - k^2) \equiv 0 \pmod 4$, where $\sigma_0(m)$ denotes the number of positive divisors of $m$, for $N = An + B$ with $(A,B) \in \{ (16,14),$ $(36,30),$ $(72,42),$ $(196,70),$ $(252,114) \}$. Our proof relies on a result of Keith which states that $\nu_2 (N) \equiv 0 \pmod 4$, where $\nu_2(N)$ is the number of partitions of $N$ with exactly two part sizes. Inspired by Dewitt and Keith, our approach combines combinatorial arguments with modular arithmetic techniques.