In this paper, we find all the solutions of the title Diophantine equation in positive integer variables $(m, n, k, x)$, where $F_i$ is the $i^{th}$ term of the Fibonacci sequence.

We study the first moments of central values of Hecke $L$-functions associated with quadratic, cubic and quartic symbols to prime moduli. This also enables usto obtain results on first moments of central values of certain families of cubic and quartic Dirichlet $L$-functions of prime moduli.

In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.

For the Minkowski question mark function $?(x)$ we consider derivative of the function \[f_n(x) = \underbrace{?(?(\ldots ?}_\text{n times}(x))).\] Apart from obvious cases (rational numbers for example) it is non-trivial to find explicit examples of numbers $x$ for which $f'_n(x)=0$. In this paper we present a set of irrational numbers, such that for every element $x_0$ of this set and for any $n\in\mathbb{Z}_+$ one has $f'_n(x_0)=0$.

In a recent paper of Dokchitser-Dokchitser-Maistret-Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first étale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture.

In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and a common area. We prove that there exist infinitely many such pairs of triangles and give several numerical examples.

In this paper we have estimated one multi-variable character sum \begin{align*} \sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\sum_{c=2}^{p-2}\sum_{d=1}^{p-1}\left(\frac{a^2-b^2}{p}\right)\left(\frac{b^2-1}{p}\right) \left(\frac{c^2-d^2}{p}\right)\left(\frac{d^2-1}{p}\right)\left(\frac{a^2c^2-1}{p}\right), \end{align*} for odd prime $p$. With the help of our estimate of the above character sum, we have studied the tenth power mean value of generalized quadratic Gauss sums using estimates for character sums and analytic methods.