相关论文: A Cylcotomic Investigation of the Fermat - Catalan…
We give a cyclotomic proof of the fact that the equation $\frac{x^p + y^p}{x+y} = p^e z^q$ has no solutions in coprime integers $x,y,z$ and $p > 3; q$, a pair of distinct odd primes.
We prove two results concerning the generalized Fermat equation $x^4+y^4=z^p$. In particular we prove that the First Case is true if $p \neq 7$.
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 paper we prove new cases of the asymptotic Fermat equation with coefficients. This is done by solving remarkable $S$-units equations and applying a method of Frey-Mazur.
We give a cyclotomic proof of the fact that the equation $\frac{x^p + y^p}{x+y} = p^e z^p$ has no solutions in coprime integers $x,y,z$ and $p > 3$, a prime. This implies in particular Fermat's Last Theorem.
We solve the diophantine equations x^4 + d y^2 = z^p for d=2 and d=3 and any prime p>349 and p>131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermat's Last Theorem, and by…
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…
We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced to the determination of the sets of rational…
In a recent paper, Freitas and Siksek proved an asypmtotic version of Fermat's Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A x^p+B y^p+ C z^p=0$, where $A$,…
In this paper we are interested in solving the Fermat-type equations x^5+y^5=dz^p where d is a positive integer and p a prime number $\ge 7$. We describe a new method based on modularity theorems which allows us to improve all the results…
This note proves two theorems regarding Fermat-type equation $x^r + y^r = dz^p$ where $r \geq 5$ is a prime. Our main result shows that, for infinitely many integers~$d$, the previous equation has no non-trivial primitive solutions such…
In this paper, we develop the modular method for the generalized Fermat equation appearing in the title, within the framework of Darmon's program and using Frey hyperelliptic curves. As an application, we study a conjecture of Laradji,…
We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific…
We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432d^3k^2$ over…
We show that the existence of a non-trivial solution of $x^n+y^n=p^n$, with $p$ a prime number, is equivalent to the existence of a solution of a certain (over-determined) system of $(n-1)$-recursion relations ("zipper" equations) in…
We prove a diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon's program. For that, we provide an…
By means of elementary conditions on coefficients, we isolate a large class of Fermat-like Diophantine equations that are not partition regular, the simplest examples being $x^n+y^m=z^k$ with $k\notin\{n,m\}$.
This article lists all the solution of the Catalan equation $x^p-y^q=1$ for $x,y \in \mathbb{Z}[i]$, when one of the primes $p and q$ is even.
We prove that the Fermat-type equation $x^3 + y^3 = z^p$ has no solutions $(a,b,c)$ satisfying $abc \ne 0$ and $\gcd(a,b,c)=1$ when $-3$ is not a square mod~$p$. This improves to approximately $0.844$ the Dirichlet density of the set of…
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.