On Distinct Residues of Factorials

= −1 for all y satisfying y(y + 4)(y + 6)− 1 ≡ 0 (mod p). He confirmed that there are no such primes less than 109. In this paper, we describe the connection between the socialist primes and the left factorial function !n = 0! + 1! + · · · + (n − 1)! introduced by Ðuro Kurepa [4]. Kurepa conjectured that gcd(!n, n!) = 2 holds for all integers n > 1, which is equivalent to the statement that there is no odd prime p that divides !p. This conjecture is also mentioned in [2, Section B44]. In our previous work [1], we calculated and recorded the residues rp=!p mod p for all primes p < 234. Now we show that if p is a socialist prime then (!p−2)2 ≡ −1 (mod p), which enabled us to immediately confirm that there are no such primes less than 234. Additionally, we extended the search up to 1011.