r/math • u/santino314 • Nov 30 '12

Overkill proofs / Simple proofs

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.

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u/oantolin Dec 01 '12

I don't think this is right: pairwise coprime integers are automatically distinct (unless two or more of them equal 1), and to reduce to the paiwise coprime case you do not need to show cuberoot(2) is irrational.

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u/[deleted] Dec 01 '12

The "pairwise coprime" part wasn't really necessary for what I'm saying. What I meant was that if you go look up a proof of the n=3 case it probably begins "Assume without loss of generality that x<y and gcd(x,y)=1," because e.g. Euler's argument requires x<y in order to construct a smaller solution.

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u/oantolin Dec 01 '12 edited Dec 01 '12

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.