Your favourite hammer proofs

[Talith]

Hopefully most undergrads understand the concept of a 'proof hammer'. That is, a theorem that's so powerful it can be used to prove a huge range of propositions, normally even when more foundational methods work just as well. I thought it might be a good idea to see what humorous/interesting hammer proofs people have come across over the years.

As this isn't too serious a thread, I don't think it's too important if the 'proof' is circular or not (for instance finding the derivative of sin(x)/x using l'hopitals rule) sometimes you get the most humorous results when the hammer proof isn't even a real proof.

Anyway to start off, here's my favourite:

Proposition: for all n>2, 2^(1/n) is irrational.

The normal proof of this propositional is similar to the proof that sqrt(2) is irrational (or you can do it using the rational root theorem). The hammer proof uses a 'heavyweight' theorem to say the least. It's interesting to note that the usual proof of the propositional also works in the n=2 case, but the hammer proof actually doesn't generalise to this easy case.

Proof: Let n>2. Suppose 2^(1/n)=p/q for some natural numbers p and q!=0, then 2=(p/q)^n=(p^n)/(q^n). Then, p^n=2q^n and so p^n=q^n+q^n. However, p and q are natural numbers and n is a natural number >2. It follows that the equality can't hold as a consequence of the Fermat-Wiles Theorem, and so the assumption that 2^(1/n)=p/q for some naturals p and q is false. Hence 2^(1/n) is irrational for all n>2.

I'm not sure if the proof is circular or not; I image the propositional is needed at some point in Wiles' proof but I haven't read it and probably wouldn't be able to follow it anyway. So, what are your favourite hammer-proofs?

[Hammer]

I am proof enough!

[pizzazz]

It's not circular, because if this fact is used in Wiles's proof, it had a different proof elsewhere. Thus every step is valid, if technically redundant. In fact, I believe there was a thread on this topic at some point (using this very example even!).

[theodds]

I used Taylor's Theorem to convince myself that for a real polynomial p(x) we have p(a) = 0 if and only if p(x) = (x - a) q(x) for some lower degree polynomial q the other day (ignoring any nitpicky conditions I'm leaving out). I can't think of anything in the development of calculus up to Taylor's theorem that needs it so I don't think it is circular.

[Proginoskes]

How about the entire book A=B, by Petkovsek, Wilf and Zeilberger? If you have a combinatorial identity to prove, the approaches in that book can be used to do a computer proof. (Whenever doing these proofs for the American Mathematical Monthly, this is always my first stop.)

See: http://www.math.upenn.edu/~wilf/AeqB.html