Let $d,N_1,N_2\in\mathbb{Z}$ with $N_2\leq N_1(N_1-1)$, $d\in\{1,\ldots,N_2\}$ and $d\mid N_2$.
Let
$p_1(d)=N_1^2-d(N_1+3)-d^2$, and
$p_2=27N_2+2N_1^3+(-18N_1-3N_1^2)d+(9-3N_1)d^2+2d^3$.
Let $F(d)=\frac{2}{3}\sqrt{p_1(d)}\cos\left[\frac{1}{3}\arccos\left[\frac{p_2(d)}{2p_1(d)^{3/2}}\right]\right]+\frac{3+N_2+d}{3}.$ Is $F$ an increasing function w.r.t. d? The standard methods get quite messy here.