Three, (15 points) Can 2010 be written as the sum of squares of $k$ distinct prime numbers? If so, determine all possible values of $k$, and provide an example for each corresponding $k$ value; if not, briefly explain the reason.

---

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
Three, (1) If 2010 can be written as the sum of squares of $k$ prime numbers, and the sum of the squares of the smallest 10 distinct prime numbers is
$$
\begin{array}{l}
4+9+25+49+121+169+ \\
289+361+529+841 \\
=2397>2010,
\end{array}
$$

Therefore, $k \leqslant 9$.
It is known that there is only one even prime number 2, and the rest of the prime numbers are odd, and the square of an odd number is congruent to 1 modulo 8.
Clearly, 2010 is not the square of a prime number, so $k \neq 1$.
The sum of the squares of two different prime numbers is congruent to 1 or 2 modulo 3, but 2010 is congruent to 0 modulo 3, so $k \neq 2$.

Since 2010 is congruent to 2 modulo 8, but the sum of the squares of $l$ (where $l=3,4,5,6,8,9$) different prime numbers is congruent to 3 or 6, 4 or 7, 5 or 0, 6 or 1, 0 or 3, 1 or 4 modulo 8, therefore,
$$
k \neq 3,4,5,6,8,9 \text {. }
$$

Notice that
$$
2010=2^{2}+3^{2}+7^{2}+11^{2}+13^{2}+17^{2}+37^{2} \text {. }
$$

Thus, $k=7$.