What is the simple proof of Fermat's Last Theorem?




There isn't yet a simple proof, that's why it took so long time to be proved, considering the statement of this Theorem is simple: For any integer $n>2$, the equation $a^n+b^n=c^n$ has no positive integer solutions, for $a,b,c$ positive integers.

It took more than $3$ centuries to have the first proof published in 1995 by the British mathematician Andrew Wiles. His proof is composed by two papers where the proof are written in $129$ pages long. This work consumed over seven years of Wiles's research time.

Wiles's proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques which were not available to Fermat. It is said that just a few mathematicians worldwide can understand this proof made by Wiles.

The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for instance, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof and it becoming known as a conjecture rather than a theorem.

So, this Theorem is known today as Fermat-Wiles Theorem, and Wiles says that his proof can be considered as a proof for the XX century for this Theorem, since the tools/resources used in his proof was not available at Fermat's time as mentioned.

Well, if someday some mathematician discover the proof that Fermat claimed he had (as he written in the margin of Diofanto's Arithmética) the mankind will be able to see if there was or not a simpler proof. For now it is a enigma.

By Francisco Constantino Simao Junior. 

Source: Fermat's Last theorem, Wikipedia.