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On the value set of n modulo a prime

We show that for infinitely many prime numbers p there are at least log log p/ log log log p distinct residue classes modulo p that are not congruent to n! for any integer n. | Turk J Math 29 (2005) , 169 – 174. ¨ ITAK ˙ c TUB On the Value Set of n! Modulo a Prime William D. Banks, Florian Luca, Igor E. Shparlinski, Henning Stichtenoth Abstract We show that for infinitely many prime numbers p there are at least log log p/ log log log p distinct residue classes modulo p that are not congruent to n! for any integer n. 1. Introduction For any odd prime p, let F (p) be the number of the distinct residue classes modulo p that are missed by the sequence {n! : n = 1, 2, . . . }. In F11 of [5], it is conjectured that F (p) ≈ p/e as p → ∞. This question appears to be quite difficult, and very little is known at the present time about the distribution of n! modulo p. Some evidence for the conjecture is provided by [1], where it is shown that for a random permutation σ of the set {1, . . . , p − 1}, the products n Y σ(i), n = 1, . . . , p − 1, i=1 hit the expected number of p(1−1/e) residue classes modulo p. It has been remarked in [3] that F (p) ≤ p−(p−1)1/2 (which is based on the simple observation that n = n!/(n−1)!). Several other results about the distribution of n! modulo p can be found in [2, 3, 4, 7, 10], but unfortunately these give very little insight into the behaviour of F (p). Here, we show that the Chebotarev Density Theorem implies that the relation lim supp→∞ F (p) = ∞ holds. Below, we give a slightly more precise form of this statement using a result from [6]. The implied constants in the symbol ‘O’ are always absolute. 169 BANKS, LUCA, SHPARLINSKI, STICHTENOTH 2. Preparations We use some standard notions of the theory of algebraic number fields which can be found in [8] and many other standard textbooks. Given two number fields K ⊂ L and a basis {β1 , . . . , β` } for L over K (thus ` = [L : K]), we denote by DL/K (β1 , . . . , β`) the discriminant of this basis. We also denote by NL/K (β) ∈ K the relative norm of an element β ∈ L. We recall the following formula for discriminants in a tower of finite extensions K ⊂