In this paper, we consider the usual Pell and Pell–Lucas sequences. The Pell sequence [Formula: see text] is given by the recurrence un = 2un-1 + un-2 with initial condition u0 = 0, u1 = 1 and its associated Pell–Lucas sequence [Formula: see text] is given by the recurrence vn = 2vn-1 + vn-2 with initial condition v0 = 2, v1 = 2. Let n, d, k, y, m be positive integers with m ≥ 2, y ≥ 2 and gcd (n, d) = 1. We prove that the only solutions of the Diophantine equation unun+d⋯un+(k-1)d = ym are given by u7 = 132 and u1u7 = 132 and the equation vnvn+d⋯vn+(k-1)d = ym has no solution. In fact, we prove a more general result.
In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation (x + 1) 3 − (x + 2) 3 + • • • − (x + 2d) 3 + (x + 2d + 1) 3 = z p , where p is prime and x, d, z are integers with 1 ≤ d ≤ 50.
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