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.
In this paper, we find all solutions of the exponential Diophantine equation +1 − = in positive integer variables ( , , ), where is the -th term of the Balancing sequence.
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 ∈ {±1, ±4} , which is a Padovan number, with a few exceptions that we completely characterize.
Let P k and Q k be the k th Pell and Pell-Lucas terms of the Pell sequence fP n g n ! 0 and the Pell-Lucas sequence fQ n g n ! 0 , respectively. In this paper, we study the Diophantine equations P n ¼ x a AE x b þ 1 and Q n ¼ x a AE x b þ 1, in positive integers (n, x, a, b) and determine the explicit upper bounds for n. We also completely solve these equations in positive integers (n, x, a, b) with 0 b\a and 2 x 20.
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