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In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and and a version of the Baker-Davenport reduction method in…

In this paper, we consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z},$ where $a, b, c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{r}, r\in Z^{+}, 2\mid r$ with $b$ even. That is $$a=\mid…

In this paper we consider Diophantine equation x4 + y4 = z4 + w4 (1)We construct some family of cubic curves.We prove that every rational point on Quar- tica x4 + y4 = z4 + w4 can be mapped to a point on some curve of this family. We also…

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x,y,m,n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability…

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative…

We implement four algorithms for solving linear Diophantine equations in the naturals: a lexicographic enumeration algorithm, a completion procedure, a graph-based algorithm, and the Slopes algorithm. As already known, the lexicographic…

Solving two-variable linear Diophantine equations has applications in many cryptographic protocols such as RSA and Elliptic curve cryptography. The Extended Euclid's algorithm is a well known algorithm to solve these equations. We revisit…

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 $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential…

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 prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, combining the Gel'fond-Baker method with an elementary approach, we prove that if $\max\{a,b,c\}>5\times 10^{27}$, then the equation $a^x+b^y=c^z$ has at…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. There is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer…

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^3+x^3+(x+1)^3=z^2$ in integers $x$, $z$. The generalization $(x-1)^k+x^k+(x+1)^k=z^n$ (with $x$, $z$, $n$ integers and $n \ge 2$) was considered by Zhongfeng…

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in…

We consider Diophantine equations of the shape $ f(x) = g(y) $, where the polynomials $ f $ and $ g $ are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many…

Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive…

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, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation…

For fixed integers $D \geq 0$ and $c \geq 3$, we demonstrate how to use $2$-adic valuation trees of sequences to analyze Diophantine equations of the form $x^2+D=2^cy$ and $x^3+D=2^cy$, for $y$ odd. Further, we show for what values $D \in…