In this paper, we find all the solutions of the title Diophantine equation in nonnegative integer variables $(m, n, x)$, where $P_k$ is the $k$th term of the Pell sequence.

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In this paper, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=k$, where $k\in\{\pm1,\pm4\}$, which is a Padovan number, with a few exceptions that we completely characterize.

In this paper, we show that there is at most one value of the positive integer XXX participating in the Pell equation X2−dY2=kX2−dY2=kX^2-dY^2=k, where k∈{±1,±4}k∈{±1,±4}k\in\{\pm1,\pm4\}, which is a Padovan number, with a few exceptions that we comp...