Related papers: There is no Diophantine $D(-1)$--quadruple
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…
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 $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers $s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if all…
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 $S$ denote a set of primes and let $a_1,\ldots,a_m$ be positive distinct integers. We call the $m$-tuple $(a_1,\ldots,a_m)$ an $S$-Diophantine tuple if $a_ia_j+1=s_{i,j}$ are $S$-integers for all $i\not=j$. In this paper, we show that…
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…
In this paper we find a parametric solution to the hitherto unsolved problem of finding three positive integers such that their sum, the sum of their squares and the sum of their cubes are simultaneously perfect squares.
We prove that for every integer $n$, there exist infinitely many $D(n)$-triples which are also $D(t)$-triples for $t\in\mathbb{Z}$ with $n\ne t$. We also prove that there are infinitely many triples with the property $D(-1)$ in…
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…
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…
For an element $r$ of a ring $R$, a Diophantine $D(r)$ $m$-tuple is an $m$-tuple $(a_1,a_2,\ldots,a_m)$ of elements of $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ is a perfect square in $R$. In this article, we compute and…
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids…
A set of m distinct positive integers {a_{1},...a_{m}} is called a Diophantine m-tuple if a_{i}a_{j}+n is a square for each 1\leqi<j\leqm . The aim of this study is to show that some P_{k} sets can not be extendible to a Diophantine…
In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields $\mathbb{F}_p$ for…
A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The problem of finding such cuboids or proving their non-existence is not solved thus far. The second…
For two relatively prime positive integers $a, b\in \mathbb{N}$, it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}$$ has a nonnegative…
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…
Even though four theorems are actually proved in this paper, two are the main ones,Teorems 1 and 3. In Theorem 1 we show that if a and be are odd squarefree positive integers satisfying certain quadratic residue conditions; then there…
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The second cuboid conjecture is one of the three propositions suggested as intermediate stages in proving the…
We show that the Diophantine pair $\{1, 3\}$ can not be extended to a Diophantine quintuple in $\mathbb{Z}\left[\sqrt{-2}\right]$. This result completes the work of the first author and establishes non-extensibility of the Diophantine pair…