Congruences via Partitions with Exactly Two Part Sizes

Abstract

We prove the congruence ${\sum }_{1\le k<\sqrt{N}}\ddot{I}{ƒ}_0(N-k^2)\equiv 0(\mathrm{mod}4)$, where $\ddot{I}{ƒ}_0(m)$ 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 $\hat{I}{½}_2(N)\equiv 0(\mathrm{mod}4)$, where $\hat{I}{½}_2(N)$ 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.

10 pages, 2 figures, 1 table