4. Let $a$ be a prime number, $b$ be a positive integer, and
$$
9(2 a+b)^{2}=509(4 a+511 b) \text {. }
$$

Find the values of $a$ and $b$.
$(2008$, National Junior High School Mathematics League)
Hint: $2a + b$ can be treated as a whole, then $9(2a + b)^2 = 509[509b + 2(2a + b)]$. It is easy to see that 509 is a prime number, so 509 and 9 are coprime.

Therefore, 509 divides $(2a + b)$.
Let $2a + b = 509k\left(k \in \mathbf{N}_{+}\right)$.
Thus, $9k^2 = b + 2k$.
So, $9k^2 = 511k - 2a$.
Hence, we can find $a = 251, b = 7$.