Rigorous Geometric Obstructions for Fourier Curves Generated by Prime Numbers

Abstract

We study planar curves defined by finite Fourier series of the form $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t}$, where the frequencies are the prime numbers and $v_p(n!)$ denotes the exponent of the prime $p$ in the factorization of $n!$. We establish several rigorous obstructions to uniform geometric regularity as $n\to\infty$. In particular, we prove that the curve lengths grow without bound, that neither the first nor the second derivatives remain uniformly bounded, and that the diameters grow at least on the order of $n\log\log n$. As a consequence, the covering numbers of the curves satisfy explicit quantitative lower bounds. These results provide a rigorous explanation for the complex geometric behavior observed in numerical investigations of this model.