nLab Fermat curve

Idea

The Fermat curve is a projective algebraic curve over a ground field $k$ given by the Fermat equation

$$x^n + y^n = z^n \,,$$

where $n$ is a positive integer greater or equal $2$.

Fermat’s last theorem is the statement, that for $n\gt 2$ this equation has no solutions in rationals (or, equivalently, has no solutions in integers), other than when at least one of $x,y$ or $z$ vanishes. In other words, the Fermat curves with $n\gt 2$ have no nontrivial $\mathbb{Q}$-rational points. For $n=2$, the Fermat curve is just the projective line $\mathbb{P}^1$, hence is much less interesting.

Famously a proof of this theorem had been announced by Pierre de Fermat, but no proof was recorded before Andrew Wiles with aid of Richard Taylor proved the modularity theorem for semistable elliptic curves, which by Ribet's theorem implies Fermat’s last theorem (see Wiles' proof of Fermat's last theorem).

Therefore, while Fermat’s theorem is generally accepted as proven now, the methods used in the proof go far beyond the mathematics known at Pierre de Fermat‘s time. Moreover Ribet's theorem and the modularity theorem are major conceptual statements in arithmetic geometry, quite different in character from the content of Fermat’s last theorem in itself:

I confess that Fermat’s Last Theorem, as an isolated proposition, has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of. (Carl Friedrich Gauss, quoted in J. R. Newman (ed.) The World of Mathematics (1956))

It is open whether there exists a more direct proof, in particular a proof using only some system of arithmetic. Harvey Friedman has gone so far as to conjecture (his ‘Grand Conjecture’) that Elementary Function Arithmetic should be enough to prove FLT, among many other ‘finitary’ major theorems in mathematics).

References

General

• wikipedia: Fermat curve, Fermat’s last theorem

• eom: Fermat’s last theorem

• Colin McLarty, What does it take to prove Fermat’s last theorem? Grothendieck and the logic of number theory, Bull. Symbolic Logic 16 (2010), no. 3, 359–377 jstor

Applications

Discussion of anomaly cancellation on the standard model of particle physics:

• Nakarin Lohitsiri, David Tong, Hypercharge Quantisation and Fermat’s Last Theorem (arXiv:1907.00514)