1975 Canadian MO Problems/Problem 7

Problem 7

A function $f(x)$ is $\textit{periodic}$ if there is a positive integer such that $f(x+p) = f(x)$ for all $x$ . For example, $\sin x$ is periodic with period $2\pi$ . Is the function $\sin(x^2)$ periodic? Prove your assertion.

Solution

To prove that $\sin(x^2)$ is periodic, we need to check if there exists a positive number $p$ such that $\sin((x + p)^2) = \sin(x^2)$ for all $x$ . Using the trigonometric property $\sin(a) = \sin(b) \iff a - b = 2n\pi$ (where $n \in \mathbb{Z}$ ), this implies $(x + p)^2 - x^2 = 2n\pi$ for some integer $n$ . Expanding and simplifying, $(x + p)^2 - x^2 = 2px + p^2$ , so the equation becomes $2px + p^2 = 2n\pi$ . Rewriting, $p(2x) + p^2 = 2n\pi$ . For this equation to hold for all $x$ , the term $p(2x)$ must vanish, which is only possible if $p = 0$ . However, since $p > 0$ is required for periodicity, no such $p$ exists, meaning $\sin(x^2)$ is not periodic. [Intuitively, the argument $x^2$ of $\sin(x^2)$ grows faster than linearly as $x$ increases, causing the values of $\sin(x^2)$ to fail to repeat in a regular pattern. Therefore, $\sin(x^2)$ is not periodic.] $\blacksquare$