Related papers: A Method of Solving a Dophantine Equation of Secon…

By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and…

The subject matter of this work is the diophantine equation x^n+y^m=c(x^k)(y^l), where n,m,k,l,c are natural numbers.We investigate this equation from the point of view of positive integer solutions.A preliminary examination of sources such…

In this paper we consider the Diophantine equation $ V_n - b^m = c $ for given integers $ b,c $ with $ b \geq 2 $, whereas $ V_n $ varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the…

In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are…

Let $\mathcal{A}$ be the set of all Diophantine equations of the form $au^2 + buv + cv^2 + du + ev + f = 0$, where $a,b,c,d,e,f \in \mathbb{Z}$ and $a > 0$. One way to solve the equation $A \in \mathcal{A}$ is by applying Lagrange's method…

In this paper we classify all monic, quartic, polynomials $d(x)\in\mathbb{Z}[x]$ for which the Pell equation $$f(x)^2-d(x)g(x)^2=1$$ has a non-trivial solution with $f(x),g(x)\in\mathbb{Z}[x]$.

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…

We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions…

In this paper, we deal with the quartic Diophantine equation $X^4-Y^4=R^2-S^2$ to present its infinitely many integer solutions.

The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer…

For every non-integral $\alpha>1$, the sequence of the integer parts of $n^{\alpha}$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with exponent $\alpha$, and let $\mathrm{PS}(\alpha)$ denote the set of all those terms. For all…

This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to…

We present a general algorithm for solving all two-variable polynomial Diophantine equations consisting of three monomials. Before this work, even the existence of an algorithm for solving the one-parameter family of equations…

In this paper we show a way to generalize the linear Diophantine equation a1x1+a2x2+...+anxn=d . We deal with the nonlinear Diophantine equation det|A X|=+-d , which generalizes the linear one, and we give a necessary and sufficient…

Motivated by the recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of…

Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…

The primary aim of this paper is to study the generalized Fermat equation \[ x^2+y^{2n} = z^{3p} \] in coprime integers $x$, $y$, and $z$, where $n \geq 2$ and $p$ is a fixed prime. Using modularity results over totally real fields and the…

The non-zero integer solution set is derived for C^n = A^n + B^n. The non-zero integer solution set for n = 2 is [C - (a + b)]^2 = 2ab. The variables a and b equal (C - A) and (C - B) respectively and are nonzero integer factors of 2M^2…

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…

Let $d\ge 2$ and $n\ge d$ with $(d,n)\notin \{(2,2),(3,3)\}$. We consider homogeneous Diophantine equations of degree $d$ in $n+1$ variables and whether they have solutions in the primes. In particular, we show that a certain local-global…