Papers › On power values of pyramidal numbers, II
On power values of pyramidal numbers, II
Andrej Dujella, Kálmán Győry, Philippe Michaud-Jacobs, Ákos Pintér
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For m ≥3, we define the $m$th order pyramidal number by Pyrₘ(x) = 1/6 x(x+1)((m-2)x+5-m). In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation Pyrₘ(x) = y² are found in positive integers x and y, for 6 ≤m ≤100. In this paper, we consider the question of higher powers, and find all solutions to the equation Pyrₘ(x) = yⁿ in positive integers x, y, and n, with n ≥3, and 5 ≤m ≤50. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.
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