Examples of revisited proofs after new theorems are discovered...

Question

Are there any nice examples of "old" complicated proofs that become much simpler after new math is discovered years later? For instance, we know now that Pn+16< Pn+1 occurs infinitely often (where Pn is the Pth prime), I guess there are many theorems about prime gaps, or many bounds in some theorems that can be revisited and add that landmark result thanks to Yitang Zhang... We also know that if the Riemann Hypothesis is true, many nice results follow, because we know of many theorems that stars with "if the RH is true..then...". Do you know of some other nice examples?

Answer 1

There is hope that techniques developed in classifying saturated fusion systems will yield results in group theory that will streamline the classification of finite simple groups.

Answer 2

Are there any nice examples of "old" complicated proofs that become much simpler after new math is discovered years later ?

Yes. Ruffini proved that quintics and higher polynomials cannot be solved by radicals in the general case, as is the case for those of inferior degree. Though there was nothing mathematically wrong with his proof, it did have the slight disadvantage that it was about $500$ pages long. Until Abel and, later, Evariste Galois, published theirs, which did not have more than a few pages each.

Answer 3

Fermat's Little Theorem was originally proved by Euler in 1736 by proving the more general case $a^{\phi(n)} \equiv 1\pmod{n}$. However, today's typical proof of Fermat's Little Theorem is proved using abstract algebra, which provides a proof simpler and more direct than Euler's proof.

Source: http://en.wikipedia.org/wiki/Fermats_little_theorem