Fermat's Last Theorem

Prerequisite: as much math as possible

Somewhere in a copy of Diophantus' Arithmetica, Fermat jotted down this. Fermat’s Last Theorem:

There is no whole number solution to xⁿ + yⁿ = zⁿ, where n > 2

Brilliant. Except that he never passed down a proof:

“I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain”

So he left mankind to suffer.

It does seem like a simple problem at first, until you realize that it actually represents an infinite amount of equations:

x³ + y³ = z³

x⁴ + y⁴ = z⁴

x⁵ + y⁵ = z⁵

...

and on and on..

Unfortunately, I do not have the mathematical knowledge to comprehend the proof. But even if I can, you probably cannot. Most people cannot. The official proof by Andrew Wiles is over a hundred pages long and contains advanced material (understatement). What I can do though, is break down the process qualitatively (and vaguely).

Pierre de Fermat

contribution: proof for n = 4 (and therefore 8, 12, 16..)

method: infinite descent (on the equation x⁴ + y⁴ = z²)

Infinite descent is a form of proof by contradiction. It first assumes the opposite to be true (that there is a solution), then feeds the hypothetical solution into a recursion. There appears to be infinitely many smaller solutions which cannot be true, since there are finitely many whole numbers in the decreasing direction. This contradiction proves that there cannot be a solution.

All n in multiples of 4 are proven since they can all be written as powers of 4.

Leonard Euler

contribution: proof for n = 3 (and therefore 6, 9, 12..)

method: infinite descent involving imaginary numbers

The proof of n = 3 is essentially an adaptation of Fermat's proof for n = 4, except that several unknowns could be filled with imaginary numbers.

Sophie Germain

contribution: proof for n = Germain Prime (2, 3, 5..)

method: proof by contradiction

A Germain prime is a prime p such that (2p +1) is also prime. Assuming that there is a solution where n = Germain Prime, either x, y, or z is a multiple of n. This assumption restricts the solutions to none at all.

Yutaka Taniyama and Goro Shimura

contribution: Taniyama-Shimura Conjecture

method: comparing Dirichlet L-series of elliptic equations and modular forms

The E-series of a particular elliptic equation tells how many solutions there are using each clock arithmetic. There is an M-series for modular forms. When these mathematicians noticed that the series matched for certain elliptic equations and modular forms, they came up with the Taniyama-Shimura Conjecture:

For every elliptic equation, there is a modular form with the same Dirichlet L-series.

Basically: every elliptic equation has an equivalent modular form.

Gerhard Frey

contribution: correlate Fermat's Last Theorem with Taniyama-Shimura Conjecture

method: rearrange xⁿ + yⁿ = zⁿ into elliptic equation

It so happens that the rearranged elliptic equation of xⁿ + yⁿ = zⁿ does not have a modular form.

If the Taniyama-Shimura Conjecture is true, then this rearranged elliptic equation should have a modular form, which it does not. Such an equation that does not have a modular form would not exist, and so this rearranged equation would not exist, and the original equation would not exist, and there would be no solution to the original equation. Then Fermat's Last Theorem would be true.

In summary: if the Taniyama-Shimura Conjecture is true, then Fermat's Last Theorem is true.

Andrew Wiles

contribution: the proof

method: proving the Taniyama-Shimura Conjecture with Galoisian group theory, Kolyvagin-Flach method, and Isawa theory

To tackle the Taniyama-Shimura Conjecture, one must handle an infinite amount of equations and their infinitely long series. Group theory allowed solutions to be condensed by common properties. As for the rest, I have no idea what is the Kolyvagin-Flach method or the Isawa theory.

Of course this post is an oversimplification. You may refer to http://fermatslasttheorem.blogspot.com/ if you like but personally, I had enough~