Consider the mathematical pendulum
$$\dot{\theta}=\omega$$ $$\dot{\omega}=-\frac{g}{L}sin\left(\theta\right)$$
How can one prove that it is impossible that the time period $T$ depends only on the length $L$, and the mass $m$, i.e, that there is no such function as $f(T,L,m)=0$.
You can directly compute the period given the largest angle $θ_\max$ as $$ T=4\int_0^{θ_\max}\frac{dθ}{\sqrt{2\frac{g}L(\cosθ-\cosθ_\max)}} =2\sqrt{\frac{L}g}\int_0^{θ_\max}\frac{dθ}{\sqrt{\sin^2(θ_\max/2)-\sin^2(θ/2)}} $$ So you get a dependence on $\frac{L}{g}$ and $θ_\max$, but not on $m$ and $L$ alone.