On quotients of a more general theorem of Wilson

Abstract: The basis of this work is a corollary and generalization of Wilson’s theorem, $(-1)^{k}k!(n-k-1)!\equiv -1\pmod{n}$ iff $n$ is non-composite, for $0\leq k\leq n-1$. This corollary generates many more quotients than those already generated by Wilson’s theorem, and we derive how they relate to each other and build on the established properties of the original quotients. The main results are expressions for sums of these quotients, modular congruences that extend the results of Lehmer, and generating functions. In addition, a solution will be provided for an open problem raised in CANT 2025 by Brian Hopkins regarding a combinatorial proof for the partition identity $p(a,3)+p(b,3)=p(c,3)$, where $a$, $b$, and $c$ comprise a Pythgagorean triple.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)