The left factorial of n is defined to be 0! + 1! + ··· + (n − 1)! and is denoted by !n. Kurepa conjectured that !n is not divisible by n for n > 2 and showed that it was sufficient to check the conjecture for odd primes p. We provide a survey of articles written on the search for a counterexample to Kurepa’s conjecture and analyse the complexity of the algorithms used. These algorithms are all linear in p; that is, exponential in logp. We develop the first known algorithm whose complexity is polynomial in logp when averaged over primes.