# Estimating cardinality of a set

Estimate |$\{<a,b,c> \in \mathbb{N}:a^2+b^2+c^3 \le n\}$| with absolute error of $O(n)$.

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Why? What have you tried? –  Henning Makholm Aug 28 '12 at 12:18
I found this in my excercise book and don't know how to manage such type of tasks. I would be grateful for any hints how to start... –  Asdf Aug 28 '12 at 12:28
@Asdf: Do you really intend $c^3$? OK with me, just less gegeometrically attractive. –  André Nicolas Aug 28 '12 at 12:48
@André Nicolas: Yes, I did, but I don't know how to solve the easier one, with $c^2$, either. –  Asdf Aug 28 '12 at 13:07
For the easier one, the volume is $\frac{4\pi}{3}n^{3/2}$. This is the number of lattice points, with error that is less than a constant times the surface area if the sphere. That gives you the $O(n)$. For $c^3$ a similar idea should work, but estimates of volume, surface area may be unpleasant. –  André Nicolas Aug 28 '12 at 13:13