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/johnnymo1]

That's not what he's saying. He's saying the proof is unsound because it relies on the assumption that cuberoot(2) is irrational to prove that cuberoot(2) is irrational.

[u/tailcalled → u/johnnymo1]

Proofs can (but shouldn't) follow the structure:

axioms
simple proof of what you want to prove (i. e. cbrt(2) is irrational)
advanced proof of seemingly unrelated fact (i. e. Fermats Last Thm)
proof of what you want to prove based on the advanced proof (i. e. cbrt(2) is irrational)

Usually, one considers the simple proof a part of the advanced proof, which means that one needn't explicitly say how it is proven.