Is there a way to rewrite this without the ceil() function? $$L = \left\lceil \frac{\sqrt{v-4 \times N}-1}{4} \right\rceil$$ This is a line in my program but I cannot get ceil() to work in GMP, so I'd like to approach this mathematically and just rewrite it so it doesn't need ceil().

You can use the floor with the difference of the desired quantity and a sufficiently large integer. In particular here we know:

$$ v > \frac{\sqrt{v-4 \times N}-1}{4} $$

Therefore:

$$ L = \left\lceil \frac{\sqrt{v-4 \times N}-1}{4} \right\rceil = v - \left\lfloor v - \frac{\sqrt{v-4 \times N}-1}{4} \right\rfloor $$

'Nuff said? There are other ways to do it, but you have $v$ on hand and this is probably the fewest additional operations (and avoids any case logic). Certainly worth a comment in the code so you don't have a long moment of confusion months down the road...