Cubic congruences and sums involving.

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