An incomplete variant of Wilson's congruence

Abstract

This article examines the nontrivial solutions of the congruence $$(p-1)\cdots(p-r) \equiv -1 \pmod p.$$ We discuss heuristics for the proportion of primes $p$ that have exactly $N$ solutions to this congruence. We supply numerical evidence in favour of these conjectures, and discuss the algorithms used in our calculations.Comment: 7 pages, 2 table