Searching for a counterexample to Kurepa’s conjecture

Abstract

Kurepa’s conjecture states that there is no odd prime p p that divides ! p = 0 ! + 1 ! + ⋯ + ( p − 1 ) ! !p=0!+1!+\cdots +(p-1)! . We search for a counterexample to this conjecture for all p > 2 34 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 + ⋯ + ( ( n − 1 ) ! ) k !^{k}n=(0!)^k +(1!)^k +\cdots +((n-1)!)^{k} , and show that for all integers 1 > k > 100 1>k>100 there exists an odd prime p p such that p ∣ ! k p p\mid !^k p .