Related papers: On Pythagorean triplets

Let $(a,b,c)$ be a primitive Pythagorean triple. Set $a=m^2-n^2$,$b=2mn$, and $c=m^2+n^2$ with $m$ and $n$ positive coprime integers, $m>n $ and $ m \not \equiv n \pmod 2$. A famous conjecture of Je\'{s}manowicz asserts that the only…

A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2$ is a square. A pythagorean pair $(a, b)$ is called a pythapotent pair of degree $h$ if there is another pythagorean pair $(k,l)$, which is not a multiple of $(a,b)$,…

How many ways can we write $n$ as a sum of $3$ positive integers, no pair of which share a common factor? We express this quantity in terms of the number of solutions to a certain class of linear Diophantine equations. This allows us to…

In this paper, we use a variety of classical and new research methods for ternary exponential Diophantine equations and extensive use of computer calculations to study the conjecture of R. Scott and R. Styer which asserts that for any fixed…

This paper investigates the exponential Diophantine equation of the form $a^x+b=c^y$, where $a, b, c$ are given positive integers with $a,c \ge 2$, and $x,y$ are positive integer unknowns. We define this form as a "Type-I transcendental…

Let $(a, b, c)$ be a primitive Pythagorean triple parameterized as $a=u^2-v^2,\ b=2uv,\ c=u^2+v^2$,\ where $u>v>0$ are co-prime and not of the same parity. In 1956, L. Je{\'s}manowicz conjectured that for any positive integer $n$, the…

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…

Diophantine quadruples are sets of four distinct positive integers such that the product of any two is one less than a square. All known examples belong to an infinite set which can be constructed recursively. Some observations on these…

A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1, \ldots, x_n)$ be a degree $d$ homogeneous form with integer coefficients. We provide sufficient conditions, similar to those…

The aim of this paper is to prove the possibility of linearization of such equations by means of introduction of new variables. For $n=2$ such a procedure is well known, when new variables are components of spinors and they are widely used…

In this paper we introduce a new diophantine equation with prime numbers. Let $[\, \cdot\,]$ be the floor function. We prove that when $1<c<\frac{23}{21}$ and $\theta>1$ is a fixed, then every sufficiently large positive integer $N$ can be…

There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…

This article reports the occurrence of binary quadratic forms in primitive Pythagorean triangles and their geometric interpretation. In addition to the well-known fact that the hypotenuse, z, of a right triangle, with sides of integral…

Some relations among Pythagorean triples are established. The main tool is a fundamental characterization of the Pythagorean triples through a chatetus which allows to determine relationships with Pythagorean triples having the same…

For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent…

In this paper we find non-negative integer solutions for exponential Diophantine equations of the type $p \cdot 3^x+ p^y=z^2,$ where $p$ is a prime number. We prove that such equation has a unique solution…

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$…

Let $\varepsilon>0$ be a small constant. In the present paper we prove that whenever $\eta$ is real and constants $\lambda _i$ satisfy some necessary conditions, then there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying…

In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where…

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$, $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$. In this article, I…