Related papers: The strong Fermat-Catalan Equation

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 show that the Fermat equation $x^p + y^p = z^p$ has no solutions in coprime positive integers $x, y, z$ for any odd prime $p$.

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.

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…

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…

We give some new, simple results on the equation X^p + Y^p = Z^q.

Within the scope of elementary number theory, we prove that, as the main result, if $1 \leq x < y < z$ are integers such that at least one of $y, z, x+y$ is prime then $x^{n}+y^{n} \neq z^{n}$ for every odd integer $n \geq 3$. This result…

In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $Ax^p+By^p=Cz^3$ with prime exponent $p$ and $A,B,C \in \mathcal{O}_K \setminus…

In this paper we prove that equations of the form $x^{13} + y^{13} = Cz^{p}$ have no non-trivial primitive solutions (a,b,c) such that $13 \nmid c$ if $p > 4992539$ for an infinite family of values for $C$. Our method consists in relating a…

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 describe a strategy to attack infinitely many Fermat-type equations of signature $(r,r,p)$, where $r \geq 7$ is a fixed prime and $p$ is a prime allowed to vary. We use a variant of the modular method over totally real subfields of…

Let $K$ be a totally real number field and $ \mathcal{O}_K$ be the ring of integers of $K$. This manuscript examines the asymptotic solutions of the Fermat equation of signature $(r, r, p)$, specifically $x^r+y^r=dz^p$ over $K$, where $r,p…

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…

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 discuss the equation $a^p + 2^\a b^p + c^p =0$ in which $a$, $b$, and $c$ are non-zero relatively prime integers, $p$ is an odd prime number, and $\a$ is a positive integer. The technique used to prove Fermat's Last Theorem shows that…

We provide the necessary conditions for the existence of solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ over any quadratic number field $K$ with $A,B,C$ pth powerfree integer numbers. We determine when $x$, $y$ and $z$ are rational numbers for…

Let $K$ be a totally real number field of odd degree. Let $l \geq 5$ be a prime with $l \nmid [K:\mathbb{Q}]$ and $\gcd(\frac{l-1}{2}, [K:\mathbb{Q}])=1$. We prove that if $2$ is inert in $K$, $l$ is non-Wieferich, i.e., $2^{l-1} \not\equiv…

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…

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…

By studying the n-dimensional cube, the author finds a new way, and use it to proves that the indefinite equation $x^n+y^n=z^n(n=>3)$ has no any solutions of positive integers.