What are some proofs that are inappropriately powerful for the problem being solved? (Something like the math equivalent of using an atom bomb to kill a spider)
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Like this: https://math.stackexchange.com/questions/648273/prove-by-induction-that-22n-1-is-divisible-by-3-whenever-n-is-a-positiv/1622941#1622941? – Batominovski Nov 29 '17 at 18:45
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4There is a whole post on that mathoverflow. – Andres Mejia Nov 29 '17 at 18:48
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1$\sqrt[3]{2}$ is irrational. Suppose not. Then there are $p,q\in\mathbb{Z}$ such that $$\sqrt[3]{2} = p/q \implies 2 = p^3/q^3 \implies q^3 + q^3 = p^3, $$ which contracticts Fermat-Wiles. – Xander Henderson Nov 29 '17 at 18:48
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Problem: Prove that, for each $n\in\mathbb{N}\setminus\{1\}$, $\sqrt[n]{2}$ is irrational.
Proof: There is a very well-known proof for the case $n=2$. Suppose that $n>2$. If there were two natural numbers $p$ and $q$ such that $\sqrt[n]2=\frac pq$, then$$p^n=2q^n=q^n+q^n.$$This is impossible by Wiles' theorem (a.k.a. Fermat's last theorem).
José Carlos Santos
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Bah! Beat me to it by 20 seconds (and in greater generality). ;) (+1) – Xander Henderson Nov 29 '17 at 18:49
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2But you have to make sure we can prove FLT without using the irrationality of $2^{1/n}$ anywhere. – Nov 29 '17 at 18:52
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1@user37208 I would be amazed if the proof used that fact, but I must admit that I did not check it. – José Carlos Santos Nov 29 '17 at 18:54
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Russell and Whitehead's proof that $1+1=2$ that took several hundreds of pages of lemma's in Principia Mathematica comes to mind.
I don't know if inappropriate is the right word in this case, since of course the very idea of P.M. was to show that you can build a lot of mathematics on top of very elementary axioms of logic and set theory, but it definitely feels like shooting a mice with a cannon.
Bram28
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