Fermat's Last Theorem

‘Cujus reī dēmōnstrātiōnem mīrābilem sānē dētēxī. Hanc marginis exiguitās nōn caperet.’ ‘I have truly discovered a wonderful proof of this fact, which this margin is too narrow to contain.’

This was a sentence that would plague mathematicians for 328 years. Penned by Pierre de Fermat shortly before his death in 1665, the ‘fact’ to which he alludes is now famous for being difficult to prove. Fermat, a lawyer, was only an amateur mathematician and had only ever published one mathematical paper - an anonymous appendix to a colleague’s work. In fact, he was rather secretive about all of his hobbyist mathematics, preferring to keep his writings to himself, and seldom writing down his formal proofs for his theorems. It wasn’t until his son, Samuel, compiled his mathematical papers for publication after his death that Fermat’s theorems became public, and the mathematicians of the time began to work at rediscovering his ‘lost’ proofs. They found all but one.

The basis of Fermat’s Last Theorem (as it has come to be known) is the concept of Pythagorean triples - sets of three positive integers, x, y, and z, such that x² + y² = z², for example, 3² + 4² = 5². Seeing that there were infinitely many Pythagorean triples, Fermat posed the question of whether similar solutions could be found for powers higher than two. What he found was that there were none; instead, he offered a conjecture that for equations of the form xⁿ + yⁿ = zⁿ, where n > 2, no non-zero integer solutions exist.

A frustratingly brief message accompanied this theorem, stating that Fermat had a "wonderful proof" but that the margin of his book was "too narrow to contain". In modern mathematics, it is widely believed that Fermat’s proof was most likely flawed in some way, much like the numerous attempts made by mathematicians who tried to retrace his path. After all, Fermat worked alone, with nobody to scrutinize his work and highlight an error. Still, there persists a hope that Fermat possessed an elegant, 17th-century solution which has continued to evade rediscovery.

Fermat’s Last Theorem has a fame that supersedes that of its creator. It has been featured in episodes of Star Trek, Doctor Who, and even The Simpsons; its appearance in the latter seems to disprove the statement. In the 1998 episode The Wizard of Evergreen Terrace, a shot of a blackboard reveals an apparent counterexample to Fermat’s Last Theorem: 3987¹² + 4365¹² = 4472¹². Actually, this is just a ‘near miss’ solution, in that the average calculator does not display enough digits to show that this statement does not hold true.

While the theorem was proved in 1993 by Andrew Wiles, it was not done in a way that would have been accessible to Fermat in the 17th century. Wiles’ proof took him seven years to formulate and it makes use of concepts such as elliptic curves, Galois representations, and modular forms, all of which were discovered in the 19th century, long after Fermat first claimed his proof. Therefore, the question remains as to whether Fermat really did have a proof for his last theorem. Either way, Fermat’s Last Theorem stands as a testament to the power of mathematical curiosity and reaffirms the elusive nature of mathematics.

Extensions:

Fermat's Last Theorem - Numberphile - https://www.youtube.com/watch?v=qiNcEguuFSA

Homer Simpson vs Pierre de Fermat - Numberphile - https://www.youtube.com/watch?v=ReOQ300AcSU

Fermat’s last theorem - MacTutor - https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat's_last_theorem/