I remember this from an old blog post of yours. There is a way to prove the theorem that primes of the form $4k+1$ are sum of two squares, and the theorem that every natural number is the sum of four squares that basically says "because $\pi>2$ for the first one, and because $\pi^2>8$ for the second one". The proofs rely on Minkowski's theorem which is a great source for $\pi$ in number theory :-), but it does sound very surprising at first.

On a different note, from analytic number theory: the average order of Euler's function $\varphi$: $$\sum_{n\le x}\varphi(n)$$ is equal to $\frac{3x^2}{\pi^2}$ "on average"