Enumeration of local and global étale algebras applied to generalized Fermat equations
Authors:   , Putz Piet Hein Casper 
Published: September 2024

In this thesis, we study a special class of Diophantine equations. Diophantine equations have been studied since ancient times. For example, the Pythagoreans (around 400 BC) already knew infinitely many integer solutions to the Diophantine equation $$x^2 + y^2 = z^2$$ (Pythagoras' equation). Around 1637, the French mathematician Pierre de Fermat wrote in the margin of his copy of emph{Arithmetica} that he found a proof of the insolvability of the Diophantine equation $$x^n + y^n = z^n$$, where $x$, $y$, and $z$ are positive integers and $nge 3$. Fermat's claim was found after his death, but without a proof. It came to be known as Fermat's last theorem, and it was only in 1995 that his theorem was finally proven by Andrew Wiles (with contributions of other mathematicians). His proof sparked a renewed interest for studying similar types of equations. This brings us to emph{generalized Fermat equations}, which are of the form $$x^q + y^r = z^s,quad ggd(x,y,z) = 1,quad x,y,z,inZZ_{>0},$$ where $q$, $r$, and $s$ are positive integers. In the last 30 years, many generalized Fermat equations have been solved. However, one group of equations remains unsolved (those with ${p,q,r}$ three distinct primes and not of the form ${2,3,p}$).

We use a method based on an idea by Henri Darmon and Andrew Granville for studying some generalized Fermat equations. Darmon and Granville proved that a generalized Fermat equation has only finitely many solutions (for fixed $q$, $r$, and $s$) if $1/q + 1/r + 1/s < 1$. However, their proof of this theorem does not give a method of also finding all solutions. Nevertheless, it is sometimes still possible to do so.

Our method consists of three steps. First, we compute the so-called emph{local 'etale algebras}. In Chapter 2, we give an algorithm for doing this using p-adic analysis. We implemented our algorithm in Magma. The second step consists of computing the emph{global 'etale algebras}. For this, we use the method of emph{Hunter searching}, which we describe in Chapter 3. The method consists of enumerating a very large (but finite) search space. Using a parallel computer program running on a supercomputer, we were able to enumerate this search space for several equations. The last step consists of reducing the global 'etale algebras to solutions of our equation using techniques in rational points on algebraic curves (for example using elliptic curve Chabauty). However, we cannot give a general method for the last step, so one is resorted to finding ad hoc solutions for each case.

Our main result is given in Chapter 4. We prove that the equation $$x^5 + y^2 = z^7,quad ggd(x,y,z) = 1,quad x,y,zinZZ_{>0}$$ has no solutions if $x$ is even or if $z$ is divisible by $4$. Moreover, we give additional restrictions on the possible solutions, and we describe some potential solutions for completely solving the equation. In Chapter 5, we prove for different choices of $a$, $b$, and $c$, that the equation $$ax^5 + by^3 = cz^{11},quad gcd(ax,by,cz) = 1,quad x,y,zinZZ_{>0}$$ has no solutions.