Take the 2-minute tour ×  I know that there are infinitely many
rational points on the (unit) circle. I am interested in the
following question:  How large has the radius of a circle to be,
such that there are at least $n$ integer (lattice) points on it?
Actually, I am not interested in the precise number, but in some
reasonable good upper bound $f(n)$. Something like, there is a
circle of radius $\leq f(n)$ that has least $n$ integer points.
share|improve this question The radius must be at least about
$\sqrt{\frac{n}{\pi}}$. Also see:
en.wikipedia.org/wiki/Gauss_circle_problem –  Dan Brumleve Oct
31 '12 at 10:13 @DanBrumleve: Sorry, but my question was
ambiguous. I meant integer points on the circle and not inside
the circle. I made the question more precise. –  A.Schulz Oct 31
'12 at 10:26 Here is a lousy bound. There are $4(e+1)$
representations of $5^e$ as a sum of two squares of integers. – 
André Nicolas Oct 31 '12 at 11:10 add comment  1 Answer  up vote
2 down vote accepted  Edit: In the following I