We determine decomposition properties of Euler polynomials and using a strong result relating polynomial decomposition and diophantine equations in two separated variables, we characterize those g(x) ∈ ℚ [x] for which the diophantine equation $-1^k+2^k-\cdots +{(-1)}^x{x}^k=g\left(y\right)$ with k ≥ 7 may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. "$d$-th root problem"), which in short asks whether the integrality of the values of the quotient (resp. $d$-th root) of two (resp. one) linear recurrences implies that this quotient (resp. $d$-th root) is itself a recurrence. We shall also relate such...

The ring of power sums is formed by complex functions on $\mathbb{N}$ of the form$$\alpha \left(n\right)=b_1{c}_1^n+b_2{c}_2^n+...+b_h{c}_h^n,$$for some $b_i\in \overline{\mathbb{Q}}$ and $c_i\in \mathbb{Z}$. Let $F(x,y)\in \overline{\mathbb{Q}}[x,y]$ be absolutely irreducible, monic and of degree at least $2$ in $y$. We consider Diophantine inequalities of the form$$\left|F(\alpha \left(n\right),y)\right|\<|\frac{\partial F}{\partial y}(\alpha \left(n\right),y)|·{\left|\alpha \left(n\right)\right|}^{-\epsilon }$$and show that all the solutions $(n,y)\in \mathbb{N}\times \mathbb{Z}$ have $y$ parametrized by some power sums in a finite set. As a consequence, we prove that the equation$$F\left(\alpha \right(n),y)=f\left(n\right),$$with $f\in \mathbb{Z}\left[x\right]$ not constant, $F$ monic in $y$ and $\alpha$ not constant, has only finitely many solutions.

A Diophantine $m$-tuple is a set of $m$ positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form $y^2=(ax+1)(bx+1)(cx+1)$, where $\{a,b,c\}$, is a Diophantine triple. In particular, we consider the elliptic curve $E_k$ defined by the equation $y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1),$ where $k\ge 2$ and $F_n$, denotes the $n$-th Fibonacci number. We prove that if the rank of $E_k\left(𝐐\right)$ is equal to one, or $k\le 50$, then all integer points on $E_k$ are given by$$\begin{array}{c}(x,y)\in \{(0\pm 1),(4F_{2k+1}{F}_{2k+2}{F}_{2k+3}\pm \left(2F_{2k+1}{F}_{2k+2}-1\right)\hfill \\ \hfill \times \left(2F_{2k+2}^2+1\right)\left(2F_{2k+2}{F}_{2k+3}+1\right)\}.\end{array}$$

In this paper, we study triples $a,b$ and $c$ of distinct positive integers such that $ab+1,ac+1$ and $bc+1$ are all three members of the same binary recurrence sequence.

We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.