Related papers: On the Exponential Diophantine Equation $(a^2-2)(b…

This work determine the entire family of positive integer solutions of the diophantine equation. The solution is described in terms of $\frac{(m-1)(m+n-2)}{2} $ or $\frac{(m-1)(m+n-1)}{2}$ positive parameters depending on $n$ even or odd.…

Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max\{a,b,c\}\geq…

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. In this…

Let $A$, $B$ be fixed positive integers such that $\min\{A,B\} > 1$, $\gcd(A,B) = 1$ and $AB \equiv 0 \bmod 2$, and let $n$ be a positive integer with $n>1$. In this paper, using a deep result on the existence of primitive divisors of Lucas…

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…

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…

Let f(n)=1 if n=1, 2^(2^(n-2)) if n \in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \in {6,7,8,...}. We conjecture that if a system T \subseteq {x_i+1=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in positive…

In this short note we present a method of solving this Diophantine equation, method which is different from Ljunggren's, Mordell's, and R.K.Guy's.

We propose a method to determine the solvability of the diophantine equation $x^2-Dy^2=n$ for the following two cases: $(1)$ $D=pq$, where $p,q\equiv 1 \mod 4$ are distinct primes with $(\frac{q}{p})=1$ and…

In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation $n!=x^2-1$. It is conjectured that this equation has only three solutions, but up to now this is an open problem. Overholt observed…

Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup…

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…

Let $m$, $n$ be positive integers such that $m>n$, $\gcd(m,n)=1$ and $m \not\equiv n \bmod 2$. In 1956, L. Je\'smanowicz \cite{Jes} conjectured that the equation $(m^2 - n^2)^x + (2mn)^y = (m^2+n^2)^z$ has only the positive integer solution…

Let $D_{1}$, $D_{2}$ be coprime odd integers with min$(D_{1},D_{2})>1$, and let $N(D_{1},D_{2})$ denote the number of positive integer solutions $(x, m, n)$ of the equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$. In this paper, we prove that…

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…

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…

We shall show that, for any positive integer $D>0$ and any primes $p_1, p_2$ not dividing $D$, the diophantine equation $x^2+D=2^s p_1^k p_2^l$ has at most $63$ integer solutions $(x, k, l, s)$ with $x, k, l\geq 0$ and $s\in \{0, 2\}$.

Let $n$ be a positive integer. We study the diophantine equation $ab(ab-1)-na=\Delta^2$, where $a,b$ are positive integers. We also show that if a system of two congruences is soluble, then an equation which is a translation of…

We prove that for each odd prime p, positive integer alpha, and non-negative integers beta and gamma, the Diophantine equation X^{2N} + 2^{2 alpha} 5^{2 beta} p^{2 gamma} = Z^5 has no solution with X, Z, N in Z^+, N > 1, and gcd(X,Z) = 1.

Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-c})$. In this paper, we consider the Diophantine equation…