Journal of the Ramanujan Mathematical Society

C. Cobeli ¹, M. Vajaitu ¹, A. Zaharescu ¹

Affiliations
1 Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, RO

Abstract

Given a prime number q and r(X) = f(X)/g(X), f, g ∈ ℤ [X], denote Ρ_r,q = {p prime : p ⩽ q, g(p) ≠ 0 (mod q)}. For r(X) not a linear polynomial we prove that the finite sequence U_r,q := { r(p) (mod q)/q: p ∈ P_r,q } becomes equidistributed mod 1 as q ⟶ ∞.

AMS (2000) Subject Classification. 11N69,11L07.

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C. Cobeli ¹, M. Vajaitu ², A. Zaharescu ¹

Affiliations
1 Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700 Bucharest, RO
2 Institute of Mathematics of the Romanian, Academy, P.O. Box 1-764, 70700 Bucharest, RO

Abstract

We investigate a problem raised by R. K. Guy concerning the distribution modulo a large prime number p of the sequence 1!, 2!,... ,p!. Numerical computations indicate that about E residue classes modulo p are missed by this sequence. We show that this is the general behavior of a randomly chosen sequence of p elements modulo p. We also show that for p large enough, the map n -> n! (mod p) is far from being a permutation of at least one of the sets {1,2,... ,p - 1} or {2,4, • • •, [p-1/4]}, in the sense that a positive proportion of residues are missed in at least one of the above sets.

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