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I'll start with an example of what I mean. Everyone is familiar with Fermat's Last Theorem: $$a^n + b^n ≠ c^n \text{for $n > 2$}$$

A while ago (while reading Gödel, Escher, Bach) I encountered the following amusing equation: $$n^a + n^b ≠ n^c \text{for $n > 2$}$$ This equation also happens to be true, and is in fact trivial to prove. It also has absolutely nothing to do with with Fermat's Last Theorem.

I'm interested in other problems of this sort. The idea is that you take a difficult problem, make a seemingly small change, and the result is a completely different (and possibly trivial) problem.

GregRos
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1 Answers1

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Fermat's Last Theorem is false in all sufficiently large finite fields $\mathbb{Z}_p$. That is, given any $m>2$, there exists a $p_0$ such that, for all primes p $\geq$ $p_0$, there are natural numbers $a, b$, and $c$ with the property that $a^m + b^m = c^m$ has a solution mod $p$. This can be proven fairly easily using some basic Ramsey theory. An astoundingly simple and short proof can be found at the end of the link below:

http://math.mit.edu/~fox/MAT307-lecture05.pdf

Kaj Hansen
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