Computing integer points on the circumference of a given circle Is there a known algorithm that efficiently outputs the set of lattice points on the circumference of a given circle? The input circle is specified using its center $(a,b)$ and its radius $r$. The inputs $a, b, $ and $r$ are not necessarily integers.
If $a=b=0$ and $r=pq$ where $p$ and $q$ are primes congruent to 1 mod 4, then from the $O(1)$ integer points on the circle one could determine the factorization of $r$. So the problem is not going to be easier than factorization.