Searching for a counterexample to Kurepa's Conjecture

TL;DR

This paper searches for counterexamples to Kurepa's conjecture up to a large bound using GPU-accelerated optimization techniques, and explores generalized versions of the conjecture.

Contribution

Introduces new optimization methods and GPU-based computation to test Kurepa's conjecture for all p<2^34, and investigates generalized factorials for k<100.

Findings

No counterexamples found for p<2^34

Existence of primes dividing generalized factorials for 1<k<100

Enhanced computational techniques for large-scale prime conjecture testing

Abstract

Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots+(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation using graphics processing units. Additionally, we consider the generalized Kurepa's left factorial given by $!^{k}n=(0!)^k +(1!)^k +\cdots+((n-1)!)^{k}$, and show that for all integers $1<k<100$ there exists an odd prime $p$ such that $p\mid !^k p$.