Cubic congruences and sums involving 3k k

Abstract

Let [Formula: see text] be a prime greater than [Formula: see text] and let [Formula: see text] be a rational [Formula: see text]-adic integer. In this paper, we try to determine [Formula: see text], and reveal the connection between cubic congruences and the sum [Formula: see text], where [Formula: see text] is the greatest integer not exceeding [Formula: see text]. Suppose that [Formula: see text] are rational [Formula: see text]-adic integers, [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we show that the number of solutions of the congruence [Formula: see text] depends only on [Formula: see text]. Let [Formula: see text] be a prime of the form [Formula: see text] and so [Formula: see text] with [Formula: see text]. When [Formula: see text] and [Formula: see text], we establish congruences for [Formula: see text] and [Formula: see text] modulo p. As a consequence, when [Formula: see text] we show that [Formula: see text] has three solutions if and only if [Formula: see text] is a cubic residue of [Formula: see text].