8. On the Cartesian plane, consider a circle of radius $R$ centered at the origin. Specify at least one value of $R$ for which exactly 32 integer points lie on such a circle (a point is called integer if its abscissa and ordinate are integers).

Hint. A natural number $x$ can be represented as the sum of the squares of two integers if and only if all prime numbers (except 2) that enter the factorization of the number $x$ to an odd power have the form $4k+1$ for some integers $k$. In particular, any prime number that gives a remainder of 1 when divided by 4 can be represented as the sum of two squares. If each of the numbers $a$ and $b$ can be represented as the sum of two squares, then the same is true for their product.
Solution. If each of the numbers $a$ and $b$ can be represented as the sum of two squares, then, as noted in the hint, their product can also be represented in such a form. Moreover, the product is usually representable as the sum of two squares in more ways than each of the factors. For example, the number 5 can be represented as the sum of two non-negative squares in only one way, up to the permutation of the summands, namely: $5=2^{2}+1^{2}$; the number 13 also in only one way: $13=2^{2}+3^{2}$, but their product already in two ways ${ }^{1}$: $65=5 \cdot 13=4^{2}+7^{2}=1^{2}+8^{2}$. By adding another prime factor that gives a remainder of 1 when divided by 4, we get 4 ways: $1105=5 \cdot 13 \cdot 17=4^{2}+$ $33^{2}=9^{2}+32^{2}=12^{2}+31^{2}=23^{2}+24^{2}$. Therefore, on the circle of radius $R=\sqrt{1105}$ in the first quadrant, there are 8 integer points:

$$
(4,33),(33,4),(9,32),(32,9),(12,31),(31,12),(23,24),(24,23)
$$

Thus, there are 32 integer points on this circle in total.

Answer: For example, $\sqrt{1105}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7f07b77cc782d9cbfea6g-4.jpg?height=66&width=1719&top_left_y=2126&top_left_x=228)
Complex numbers. Notice that $5=(2+i)(2-i)=2^{2}+1^{2}$. The number 5 is equal to the sum of two squares because it can be represented as the product of two conjugate complex (Gaussian) numbers. Similarly, $13=(2+3 i)(2-3 i)=2^{2}+3^{2}$. At the same time, the number $65=5 \cdot 13$ can be represented as the product of two conjugate complex factors in two ways:

$$
\begin{gathered}
5 \cdot 13=(2+3 i)(2+i) \cdot(2-3 i)(2-i)=(1+8 i) \cdot(1-8 i)=1^{2}+8^{2} \\
\text { or } \\
5 \cdot 13=(2-3 i)(2+i) \cdot(2+3 i)(2-i)=(7-4 i) \cdot(7+4 i)=7^{2}+4^{2} .
\end{gathered}
$$