Related papers: Unit equations and Fermat surfaces in positive cha…

Let $K$ be a number field, and $S$ a finite set of places in $K$ containing all infinite places. We present an implementation for solving the $S$-unit equation $x + y = 1$, $x,y \in\mathscr{O}_{K,S}^\times$ in the computer algebra package…

We study one-parameter families of S-unit equations of the form f(t)u+g(t)v=h(t), where f, g, and h are univariate polynomials over a number field, t is an S-integer, and u and v are S-units. For many possible choices of f, g, and h, we are…

We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.

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 study ``forms of the Fermat equation'' over an arbitrary field $k$, i.e. homogenous equations of degree $m$ in $n$ unknowns that can be transformed into the Fermat equation $X_1^m+...+X_n^m$ by a suitable linear change of variables over…

We solve the Fermat-type equation \[ x^{13} + y^{13} = 3 z^7, \qquad \gcd(x,y,z) = 1 \] combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and…

We consider three types of isogonal conjugacy of two points with respect to a given triangle and characterize any of these types by a geometric equality. As an application to the Fermat problem with positive weights, we prove that in the…

[This is an older version of the paper, which will be updated soon.] In the present paper, we continue our research on the generalized Fermat equation $x^r + y^s = z^t$ with signature $(r, s, t)$, where $r, s, t \ge 2$ are positive integers…

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…

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…

Let K be an arithmetic function field, that is, a field of finite type over the rational number field. In this note, as an application of the height theory due to Chen-Moriwaki, we would like to show that the solutions of Fermat's curve X^N…

For any fixed coprime positive integers $a,b$ and $c$ with $\min\{a,b,c\}>1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x,y$ and $z$, except for one specific case which exactly gives three…

In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $$ x^n + y^n = z^5. $$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying…

Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit…

Let $\alpha,\beta,\gamma\in\mathbb{N}$. We prove that given an $r$-colouring of $\mathbb{F}_p$ with $p$ prime, there are more than $c_{r,\alpha,\beta,\gamma} p^2$ solutions to the equation $x^\alpha+y^\beta=z^\gamma$ with all of $x,y,z$ of…

In this paper, by making use of properties of elliptic functions, we describe meromorphic solutions of Fermat-type functional equations $f(z)^{n}+f(L(z))^{m}=1$ over the complex plane $\mathbb{C}$, where $L(z)$ is a nonconstant entire…

In this paper we improve upon in terms of S the best known effective upper bounds for the solutions of S-unit equations and decomposable form equations.

In the present paper we give the general solution of the functional equation $$ (f(x)+g(y)+h(z))^2 = F(x)+G(y)+H(z), x+y+z=0 $$ which is related to the exact factorized ground-state wave function for the quantum one-dimensional problem of…

In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using…

We show how to effectively solve 5-term $S$-unit equations when the set of primes $S$ has cardinality at most 3, and use this to provide an explicit answer to an old question of D.J. Newman on representations of integers as sums of…