Related papers: A solution to a problem and the Diophantine equati…

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 use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this result, we can also solve the diophantine equation in the…

In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are…

In this paper we consider the Diophantine equation $x^2+q^{2m}=2y^p$ where $m,p,q,x,y$ are integer unknowns with $m>0,$ $p$ and $q$ are odd primes and $\gcd(x,y)=1.$ We prove that there are only finitely many solutions $(m,p,q,x,y)$ for…

In this paper we investigate Diophantine equations of the form $T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if…

In this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r \in \mathbb{N}$. Further,…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

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…

We construct integer solutions {a,b} to the coupled system of diophantine quadratic-cubic equations a^2+b^2=x^3 and a^3+b^3=y^2 for fixed ratios a/b.

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…

The title equation is completely solved in integers $(n,x,y,a,b)$, where $n\geq 3$, $\gcd(x,y)=1$ and $a,b\geq 0$. The most difficult stage of the resolution is the explicit resolution of a quintic Thue-Mahler equation. Since it is for the…

Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique.…

In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title for all pairs (a,b) such that 1<a<b<101 with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation in the…

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 $n$ be a positive integer. We discuss pairs of distinct odd primes $p$ and $q$ not dividing $n$ for which the Diophantine equations $pq=x^2+ny^2$ have integer solutions in $x$ and $y$. As its examples we classify all such pairs of $p$…

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.…

Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $\alpha$ with $\alpha>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n…

We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\geq 1, m\geq 0$ and $n\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\not\equiv 3 \pmod 4$. We first solve this equation for…

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. We develop…

Let $f(x)=x^{2}(x^{2}-1)(x^{2}-2)(x^{2}-3).$ We prove that the Diophantine equation $ f(x)=2f(y)$ has no solutions in positive integers $x$ and $y$, except $(x, y)=(1, 1)$.