Why Number Theory is Cool

I’ve started doing an intro to number theory, and one of the cool things about it is that it examines some simple questions with complicated answers. Here’s a quick example:

Show that $$n^2 – n$$ is even, for any given integer n.

And here’s the proof: we can easily see that $$n^2 – n = n(n-1)$$. Since either $$n$$ or $$n-1$$ must be even, and the product of an even number with any other number is even, their product must also be even.

This sort of problem has a property similar to many geometry problems, which is that the original statement is easily comprehensible to people without a mathematical background. In this case, the solution also is widely accessible. There are some more complex proofs, though, and I’ve put one below the fold if you’re interested.
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