Related papers: Rational Diophantine sextuples with strong pair

A finitely generated subgroup {\Gamma} of a real Lie group G is said to be Diophantine if there is \beta > 0 such that non-trivial elements in the word ball B_\Gamma(n) centered at the identity never approach the identity of G closer than…

We show that a finite type duality group of dimension $d>2$ is the fundamental group of a $(d+3)$-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some…

Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper we study Diophantine graphs, i.e., finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only…

For non-zero integers $n$ and $k\geq2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $S = \{a_1,a_2,\ldots, a_m\}$ such that $a_ia_j + n$ is a $k$-th power for $1\leq i< j\leq m$. Define…

In the classical sense, the set B consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field Q(i). G.J. Rieger (1965) and…

We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3.…

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.

Let $k \geq 2$, $q$ be an odd prime power, and $F \in \mathbb{F}_q[x_1, \ldots, x_k]$ be a polynomial. An $F$-Diophantine set over a finite field $\mathbb{F}_q$ is a set $A \subset \mathbb{F}_q^*$ such that $F(a_1, a_2, \ldots, a_k)$ is a…

Let $A,B,C,D$ be rational numbers such that $ABC \neq 0$, and let $n_1>n_2>n_3>0$ be positive integers. We solve the equation $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = f(g(x)),$$ in $f,g \in \mathbb{Q}[x]$. In sequel we use Bilu-Tichy method to…

Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive…

Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of…

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $\sigma_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for…

Let n be a nonzero integer and a_1 < a_2 < ... <a_m positive integers such that a_i*a_j + n is a perfect square for all 1 <= i < j <= m. It is known that m <= 5 for n = 1. In this paper we prove that m <= 31 for |n| <= 400 and m < 15.476…

In this paper, we give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine $m$-tuples with the property $D(4)$. We also confirm the conjecture of the uniqueness of such an extension in some special…

We prove several results about the behavior Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the non-pair setting.…

Motivated by the theory of Diophantine $m$-tuples, we study rational points on quadratic twists $H^d:d y^2=(x^2+6x-18)(-x^2+2x+2)$, where $|d|$ is a prime. If we denote by $S(X)=\{ d \in \mathbb{Z}: H^d(\mathbb{Q})\ne \emptyset, |d|…

Except for a limited number of cases, a complete classification of the Diophantine sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets…

In a simple integer chain, if $u_{i-1}$, $u_i$, and $u_{i+1}$ are three consecutive terms of the chain, and the pair $(u_{i-1}, u_i)$ has a certain property, then the next pair $(u_i, u_{i+1})$ also has the same property. We extend the idea…

Recently, Hajdu and S\'{a}rk\"{o}zy studied the multiplicative decompositions of polynomial sequences. In particular, they showed that when $k \geq 3$, each infinite subset of $\{x^k+1: x \in \mathbb{N}\}$ is multiplicatively irreducible.…

We give a structure theorem for inside factorial domains. As an example we study the monoid of nonnegative integer solutions of equations of the form $a_1x_1+\cdots +a_{r-1}x_{r-1}=a_rx_r$, with $a_1,\ldots,a_r$ positive integers. This set…