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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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At the beginning of the question you seem to fix $d$ (think for example what happens if $N_2$ is prime). So, in what sense is $F(d)$ a function? –  Martin Argerami Feb 11 '12 at 1:56
    
Ca you give some background? To me it is not obvious at all if this is interesting or not. –  AD. Feb 12 '12 at 6:09
    
exact duplicate math.stackexchange.com/questions/51645/increasing-function –  Looft Feb 2 '13 at 20:04

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