Solution
Lattice Points on a Circle
Show the problem again
How many points $(x, y)$ with integer coordinates (positive, negative or zero) lie on the circle $x^2 + y^2 = 325$? Give an integer.
Worked solution
The answer is 24. By Jacobi's two-square theorem, the number of integer solutions of $x^2 + y^2 = n$ is $4(d_1(n) - d_3(n))$, where $d_1$ and $d_3$ count divisors congruent to 1 and 3 mod 4. Here $325 = 5^2 \cdot 13$, whose divisors 1, 5, 13, 25, 65, 325 are all $\equiv 1 \pmod 4$. So there are $4 \times 6 = 24$ points. Explicitly, $325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2$, and each representation with $0 < a < b$ yields 8 points via signs and swapping: $3 \times 8 = 24$.
Source: Counting representations as sums of two squares via the Jacobi two-square theorem; standard number theory. Statement written for AxiomIQ.