When taking a regular planar polygon of $M$ sides and length $2\pi $ as the initial datum of the vortex filament equation, $\mathbf{X}_{t}= \mathbf{X}_{s}\wedge \mathbf{X}_{ss}$ , the solution becomes polygonal at times of the form $t_{pq} = (p/q)(2\pi /M^{2})$ , with $\gcd (p,q)=1$ , and the corresponding polygon has $Mq$ sides, if $q$ is odd, and $Mq/2$ sides, if $q$ is even. Moreover, that polygon is skew (except when $q = 1$ or $q = 2$ , where the initial shape is recovered), and the angle $\rho $ between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time $t_{pq}$ , $\cos ^{q}(\rho /2) = \cos (\pi /M)$ , if $q$ is odd, and $\cos ^{q}(\rho /2) = \cos ^{2}(\pi /M)$ , if $q$ is even. Since the transition of one side of the polygon to the next one is given by a rotation in $\mathbb{R}^{3}$ determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle $2\pi /M$ , which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.
