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 find all solutions of the exponential Diophantine equation $B_{n+1}^x-B_n^x=B_m$ in positive integer variables $(m, n, x)$, where $B_k$ is the $k$-th term of the Balancing sequence.

Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors.

Let $p$ be an odd prime. In this paper, we consider the equation $x^{2}+p^{2m}=2y^{n},~\gcd(x,y)=1,n>2$ and we give a description for all its solutions. Moreover, we prove that this equation has no solution $(x,y,m,n)$ such that $n>3$ is an odd prime...

Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors.

Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors.

Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors.

Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors.

Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors.