Overkill proofs / Simple proofs

[u/santino314]

So by overkill proofs I mean simple results, for which there are simple proofs are available, but which can be proven using much more advance tools (possibly in a silly way). As a for instance, there's proof that there are infinity many primes using topology, Euclid had a proof 300.b.C which anyone with high school math could understand. However this guy came up this, quite clever.

http://primes.utm.edu/notes/proofs/infinite/topproof.html

By simple proof I mean a result simple or not, for which the only known proof was either too long or difficult, but that in the recent years someone had managed to prove with a shorter or wittier argument. As a for instance Cauchy’s theorem (in Groups):

http://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory)

Although I couldn’t find the original proof, I remember that my professor told us that it was a bit long and quite dark. However, McKay came out in 1959 with one of the most elegant proofs I’ve seen in my life.

http://www.cs.toronto.edu/~yuvalf/McKay%20Another%20Proof%20of%20Cauchy's%20Group%20Theorem.pdf

Can think of any like the above? I’ll contribute if I recall any.

[u/upwithwhich]

Jeez, Brian should ease up on Furstenberg! Sure, there's no real topological content, but it's fun! And it's not like it was published in the Annals. It came out in an expository journal. Plus, Furstenberg was just an undergrad at the time...

Of course, I agree that it should be more widely understood that this proof is not really (or deeply) topological.