A Superficial Proof of Fermat’s Last Theorem

I mentioned my interest in learning about Fermat's Last Theorem in an earlier post. As I wrote in that post, the theorem is surprisingly simple to state but the proof required very sophisticated mathematics. Let us recall the theorem here:

Fermat's Last Theorem: For any integer $n>2$ , the equation $x^n+y^n=z^n$ does not have non-zero integer solutions in $x$ , $y$ and $z$ .

The proof essentially draws on the following two conjectures. (I will not define the terms below in this post, as I have not reached the stage to understand the terms completely.)

Taniyama-Shimura Conjecture: Every rational elliptic curve is modular.

Frey's Conjecture: If $a$ , $b$ and $c$ are non-zero integers satisfying the equation $x^n+y^n=z^n$ , then the elliptic curve $y^2=x(x-a^n)(x+b^n)$ is not modular.

Note that these two conjectures combine to imply Fermat's Last Theorem. Indeed, by contradiction, suppose Fermat's equation has a non-zero integer solution (i.e. Fermat's Last Theorem is false). Frey's Conjecture then shows that the elliptic curve $y^2=x(x-a^n)(x+b^n)$ is not modular. However, Taniyama-Shimura implies that it should be modular. This contradiction shows that Fermat's Last Theorem cannot have a non-zero integer after all.

Frey's Conjecture was resolved in 1986 when Ken Ribet proved the so-called $\epsilon$ -conjecture which implied Frey's Conjecture. After learning this fact, Andrew Wiles proved the Taniyama-Shimura conjecture for a class of elliptic curves called semistable curves (to which the elliptic curve in Frey's conjecture belongs). By the above reasoning, Andrew Wiles thus provided the final piece to settle Fermat's Last Theorem.