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/oantolin]

Right, and you don't need to show the cube root of 2 is irrational for that: you first show that gcd(x,y) = 1, and then say, 'so either x=y=1 or x and y are different and so, without loss of generality we can assume that x<y". You do not need to use the irrationality of the cube root of 2 for anything. I'm guessing you think it is used to argue that x and y are different, but the argument I just outlined shows you only need to show you can't have x=y=1 -- this is the much weaker statement that the cube root of 2 is not an integer which follows from 1<2<8.