How to prove that $x(t) = cos{(frac{pi}{8}cdot t^2)}$ aperiodic?

Question

How to prove that $x(t) = cos{(frac{pi}{8}cdot t^2)}$ aperiodic?
My process was as follows:
$x(t+T)= cos{(frac{pi(t+T)^2}{8})}$.
So, $T^2 + 2tT -16=0$ which seems periodic to me...
Can someone tell me how to prove it?

J.G. · Accepted Answer

Another proof by contradiction: since $cosfrac{pi T^2}{8}=1$, some $ninBbb N$ satisfies $T=4sqrt{n}$, whence$$-1=cosfrac{pi(T+pi)^2}{8}=cosfrac{pi(16n+8pisqrt{n}+pi^2)}{8}implies2n+pisqrt{n}+frac{pi^2}{8}inBbb Zsetminus2Bbb Z.$$This contradicts the fact that $pi$ is transcendental.

Correct answer by J.G. on February 16, 2021

Bastien Tourand · Answer

Suppose that $x$ periodic.
Then its derivative wrt $t$ is also.
However $x'(t)=-frac{1}{4} pi x sin(frac{pi}{8}x^2)$, which is obviously aperiodic because of the factor $x$.

How to prove that $x(t) = cos{(frac{pi}{8}cdot t^2)}$ aperiodic?

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