相关论文: On a class of generalized Fermat equations of sign…

Let $n$ be a positive integer and consider the Diophantine equation of generalized Fermat type $x^2+y^{2n}=z^3$ in nonzero coprime integer unknowns $x,y,z$. Using methods of modular forms and Galois representations for approaching…

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…

In this paper, we give all the solutions of the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n, in nonnegative integers x, y, n>=3 with x and y coprime, except for the case when alpha.x is odd and beta is even.

In this paper, we gave solutions of the Diophantine equations 16^{x}+p^{y}=z^{2}, 64^{x}+p^{y}=z^{2} where p is an odd prime, n is a positive integer and x,y,z are non-negative integers. Finally we gave a generalization of the Diophantine…

We give a method to solve generalized Fermat equations of type $x^4 + y^4 = q z^p$, for some prime values of $q$ and every prime $p$ bigger than 13. We illustrate the method by proving that there are no solutions for $q= 73, 89$ and 113.

In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we…

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 title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there are…

The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar Diophantine equation p u^2 =…

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.

Suppose that $x$ is odd, $n\geq7$ and $p\notin\{2,5\}$ are primes. In this paper, we prove that the Diophantine equations $x^{2}\pm5^{\alpha}p^{n}=y^{n}$ have no solutions in positive integers $\alpha,x,y$ with $gcd(x,y)=1$.

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on…

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…

We solve the Diophantine equations $x^5 + y^5 = dz^p$ with $d=2, 3$ for a set of prime numbers of density 1/4, 1/2, respectively. The method consists in relating a possible solution to another Diophantine equation and solving the later by…

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 paper, we summarize the work on ternary Diophantine equation of the form $Ax^n+By^n=cz^m$, where $m \in \{2,3,n\} $, $n\geqslant 7 $ is a prime. Moreover, we completely solve some particular cases ($A=5^{\alpha}, ~B=64,~ c=3, ~m=2;…

Fix a rational prime $r \geq 5$. In this article, we study the integer solutions of the generalized Fermat equation of signature $(2p,2q,r)$, namely $x^{2p}+y^{2q}=z^r$, where the primes $p,q \geq 5$ are varying. For each rational prime $r…

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4, 6$ and each choice of the signs our equation has infinitely many co-prime positive…

In this article we study solutions to the generalized Fermat equation $x^q+y^p+z^r=0 $ using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at…

Given odd, coprime integers $a$, $b$ ($a>0$), we consider the Diophantine equation $ax^2+b^{2l}=4y^n$, $x, y\in\Bbb Z$, $l \in \Bbb N$, $n$ odd prime, $\gcd(x,y)=1$. We completely solve the above Diophantine equation for…