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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.

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    You're right. We should probably close the question and ban this user/IP address. /.2011-07-15

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A hint: Forget about $N_1$, $N_2$, $d$ being integers. You only have a parameter $p \geq -{1\over4}$ replacing $N_1$ and a variable $x$ with $1\leq x\leq p^2-p$ replacing your $d$. Now plot your function $F$ as a function of $x=d$ for various values of $p$ and look at the resulting figures. If you like what you see try to prove the corresponding conjectures by analytical means.

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    Anything new? Thanks.2011-07-21