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 prove that Hilbert’s Tenth Problem for a ring of integers in a number field $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over $K$). We relate division-ample sets to arithmetic of abelian varieties.

We note that every positive integer N has a representation as a sum of distinct members of the sequence ${d(n!)}_{n\ge 1}$, where d(m) is the number of divisors of m. When N is a member of a binary recurrence $u={uₙ}_{n\ge 1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

Soient $A,B,C=A+B$ trois éléments de l’ensemble ${\mathbb{N}}^{*}$ des entiers > $0$ (resp. $ℂ\left[X\right]$) des polynômes complexes) premiers entre eux ; on note $r\left(ABC\right)$ le produit des facteurs premiers (resp. le nombre des facteurs premiers dans $ℂ\left[X\right]$) du produit $ABC$. La conjecture $\left(abc\right)$ énonce que, pour tout $ϵ\>0$, il existe $C_{ϵ}\>0$ pour lequel l’inégalité : $r\left(ABC\right)\ge C_{ϵ}{S}^{1-ϵ}$ avec $S=$ max$(A,B,C)$) est toujours vérifiée. Le théorème de Mason établit l’inégalité, $D$ (supposé > $0$) désignant le plus grand des degrés des polynômes $A,B,C:r\left(ABC\right)\ge D+1$. Les cas de triplets de polynômes où l’égalité...

Let ${\left(G_n\right)}_{n\ge 1}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are $\left\{U_n\right\}$ and $\left\{V_n\right\}$, respectively. We show that the Diophantine equation $G_n=B·(g^{lm}-1)/(g^l-1)$ has only finitely many solutions in $n,m\in {\mathbb{Z}}^{+}$, where $g\ge 2$, $l$ is even and $1\le B\le g^l-1$. Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on...

In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for $(A,B,C)$ of the Diophantine equation $-(2^p·A^6)+B^3=C^2$ for any positive integral values of $p$ when $p\equiv 1$ (mod 6) or $p\equiv 2$ (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.

We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.