The Generalised Fermat Equation

On the Zeta Function of Forms of Fermat Equations

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 Xm 1 +. . .+Xm n by a suitable linear change of variables over an algebraic closure of k. Using the method of Galois descent, we classify all such forms. In the case that k is a finite field of characteristic greater than m that ...

Faltings plus Epsilon, Wiles plus Epsilon, and the Generalized Fermat Equation

Wiles’ proof of Fermat’s Last Theorem puts to rest one of the most famous unsolved problems in mathematics, a question that has been a wellspring for much of modern algebraic number theory. While celebrating Wiles’ achievement, one also feels a twinge of regret at Fermat’s demise. Is the Holy Grail of number theorists to become a mere footnote in the history books? Hoping to keep some of the sp...

A Variant of Fermat ’ S Equation 3

Modular congruences, Q-curves, and the diophantine equation x 4 + y 4 = z p , preprint; available at: http://www.math.leidenuniv.nl/gtem/view.php (preprint 55

We prove two results concerning the generalized Fermat equation x 4 + y4 = z. In particular we prove that the First Case is true if p 6= 7.

The Diophantine Equation x4 ± y4 = iz2 in Gaussian Integers

The Diophantine equation x4 ± y4 = z2, where x, y and z are integers was studied by Fermat, who proved that there exist no nontrivial solutions. Fermat proved this using the infinite descent method, proving that if a solution can be found, then there exists a smaller solution (see for example [1], Proposition 6.5.3). This was the first particular case proven of Fermat’s Last Theorem (which was ...