Abstract
Let p be a prime greater than 3. In this paper, by using expansions and congruences for Lucas sequences and the theory of cubic residues and cubic congruences, we establish some congruences for [Formula: see text] and [Formula: see text] modulo p, where [x] is the greatest integer not exceeding x, and m is a rational p-adic integer with m ≢ 0 ( mod p).
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