Related papers: Diophantine $D(n)$-quadruples in $\mathbb{Z}[\sqrt…

In this paper, first, we prove that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\equiv 0,1,4\pmod{5}$. Second, we show that the above…

We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…

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…

Let $k\geq 2$ and $n\neq 0$. A Diophantine tuple with property $D_k(n)$ is a set of positive integers $A$ such that $ab+n$ is a $k$-th power for all $a,b\in A$ with $a\neq b$. Such generalizations of classical Diophantine tuples have been…

Let A(n) be a $k\times s$ matrix and $m(n)$ be a $k$ dimensional vector, where all entries of A(n) and $m(n)$ are integer-valued polynomials in $n$. Suppose that $$t(m(n)|A(n))=#\{x\in\mathbb{Z}_{+}^{s}\mid A(n)x=m(n)\}$$ is finite for each…

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(\lambda)+1/y(\lambda)+1/z(\lambda)$ with $n=b+a\lambda$ are explicitly given for a,b coprime and a not a multiple of m . The…

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy+n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

We prove that there exist infinitely many quartic rational Diophantine quadruples, that is, sets of four pairwise distinct nonzero rational numbers whose pairwise products increased by 1 are fourth powers in Q. To the best of our knowledge,…

In this short note we study the existence and number of solutions in the set of integers ($Z$) and in the set of natural numbers ($N$) of Diopahntine Equations of second degree with two variables of the general form $ax^2-by^2=c$.

In this paper we provide criteria for the insolvability of the Diophantine equation $x^2+D=y^n$. This result is then used to determine the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$. We also determine some criteria for the…

Let $(a_1,\dots, a_m)$ be an $m$-tuple of positive, pairwise distinct, integers. If for all $1\leq i< j \leq m$ the prime divisors of $a_ia_j+1$ come from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In this note we…

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…

In this paper, we gave solutions of the Diophantine equations 16^{x}+p^{y}=z^{2}, 64^{x}+p^{y}=z^{2} where p is an odd prime, n is a positive integer and x,y,z are non-negative integers. Finally we gave a generalization of the Diophantine…

A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm…

Ramanujan found five solutions to the exponential Diophantine equation $z^2+7= 2^n$ where $z$ and $n$ are positive integers, and posed the problem of determining whether there are any more. Nagell was the first to prove that there were not.…

We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the integer…

We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation \[ n^x=(n+1)^y+(n+2)^z+(n+3)^w \] in positive integers $n,x,y,z$ and $w$ are given by…

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type…

A set of $m$ distinct nonzero rationals $\{a_1, a_2,\ldots, a_m\}$ such that $a_i a_j+1$ is a perfect square for all $1\le i <j \le m$, is called a rational Diophantine $m$-tuple. If in addition, $a_i^2+1$ is a perfect square for $1\le i\le…