Exponents 3 and 4 of Fermat’s Last Theorem and the Parametrisation of Pythagorean Triples

This document gives a formal proof of the cases n = 3 and n = 4 (and all their multiples) of Fermat’s Last Theorem: if n > 2 then for all integers x, y, z: x + y = z =⇒ xyz = 0. Both proofs only use facts about the integers and are developed along the lines of the standard proofs (see, for example, sections 1 and 2 of the book by Edwards [Edw77]). First, the framework of ‘infinite descent’ is being formalised and in both proofs there is a central role for the lemma gcd(a, b) = 1 ∧ ab = c =⇒ ∃ k : |a| = k. Furthermore, the proof of the case n = 4 uses a parametrisation of the Pythagorean triples. The proof of the case n = 3 contains a study of the quadratic form x + 3y. This study is completed with a result on which prime numbers can be written as x + 3y. The case n = 4 of FLT, in contrast to the case n = 3, has already been formalised (in the proof assistant Coq) [DM05]. The parametrisation of the Pythagorean Triples can be found as number 23 on the list of ‘top 100 mathematical theorems’ [Wie]. This research is part of an M.Sc. thesis under supervision of Jaap Top and Wim H. Hesselink (RU Groningen). The author wants to thank Clemens Ballarin (TU München) and Freek Wiedijk (RU Nijmegen) for their support. For more information see [Oos07].