Related papers: Extendibility of Some P_{k} Sets

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are…

Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers ${a_1, a_2, . . . , a_m}$ such that $a_ia_j + q$ is a perfect square for all $1 \leq i < j \leq m$. By counting integer solutions $x \in [1, b]$ of…

We study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$. Firstly, we show that it can be extended to a $D(4)$-triple with an element c, which is a member of a family of positive integers…

A set of positive integers with the property that the product of any two of them is the successor of a perfect square is called Diophantine $D(-1)$--set. Such objects are usually studied via a system of generalized Pell equations naturally…

This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(\alpha)$, where $S(\alpha)$ is the set of all integers of the form $\lfloor…

We consider Diophantine quintuples $\{a, b, c, d, e\}$, sets of distinct positive integers the product of any two elements of which is one less than a perfect square. Triples of the first kind are the subsets $\{a, b, d\}$ with $d> b^{5}$.…

In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>3, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two…

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are…

Let k>=2 and let (Q_{n}^{(k)})_{n>=2-k} be the k-generalized Pell sequence defined by Q_{n}^{(k)}=2Q_{n-1}^{(k)}+Q_{n-2}^{(k)}+...+Q_{n-k}^{(k)} for n>=2 with initial conditions Q_{-(k-2)}^{(k)}=Q_{-(k-3)}^{(k)}=...=Q_{-1}^{(k)}=0,…

Suppose $n$ is the fundamental discriminant associated with a quadratic extension of $\mathbb{Q}$. We show that for every Diophantine $m$-tuple $ \{t_1, t_2, \ldots, t_m\} $ with the property $ D(n) $, there exists integral ideals $…

In this paper, we prove that the equation $x^2-(p^{2k+2}+1)y^2=-p^{2l+1}$, $l \in \{0,1,\dots,k\}, k \geq 0$, where $p$ is an odd prime number, is not solvable in positive integers $x$ and $y$. By combining that result with other known…

In this paper, we examine the Diophantine problem given by the equation $F_n = F_l^k (F_l^m - 1)$, where $n, l, m \geq 1$ and $k \geq 3$. Here, $\{ F_t \}_{t=0}^{\infty} $ denotes the Fibonacci numbers, defined by the recurrence relation…

For $m=3,4,\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\binom n2+n\ (n=0,1,2,\ldots)$. For positive integers $a,b,c$ and $i,j,k\ge3$ with $\max\{i,j,k\}\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for…

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial…

Let $k$ be a positive integer. In this paper, we prove that if $\{k,k+1,c,d\}$ is a $D(-k)$-quadruple with $c>1$, then $d=1$.

In this paper, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main…

In this paper, we study properties of the Diophantine exponents $w_n$ and $w_n^{*}$ for Laurent series over a finite field. We prove that for an integer $n\geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of…

A Diophantine $m$-tuple with elements in the field $K$ is a set of $m$ non-zero (distinct) elements of $K$ with the property that the product of any two distinct elements is one less than a square in $K$. Let $X: (x^2-1)(y^2-1)(z^2-1)=k^2,$…

Let $C_n=n2^n+1$ denote the $n$th Cullen number. There has been recent interest in finding all Cullen numbers having a given Diophantine property. We prove that, for a fixed integer $k$ and bounded integers $a_1,\ldots,a_k$, the greatest…

We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \leq…