Three. (25 points) 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 $b$.
Three, Equation (1) is $\left(\frac{6 a+3 b}{509}\right)^{2}=\frac{4 a+511 b}{509}$.
Let $m=\frac{6 a+3 b}{509}, n=\frac{4 a+511 b}{509}$. Then
$$
b=\frac{509 m-6 a}{3}=\frac{509 n-4 a}{511} \text {. }
$$

Thus, $3 n-511 m+6 a=0$.
Since $n=m^{2}$, we have
$$
3 m^{2}-511 m+6 a=0 .
$$

From Equation (1), we know that $(2 a+b)^{2}$ is divisible by 509, and since 509 is a prime number, $2 a+b$ is divisible by 509. Therefore, $m$ is an integer, meaning the quadratic equation (3) in $m$ has integer roots. Thus, its discriminant $\Delta=511^{2}-72 a$ is a perfect square.

Let $\Delta=511^{2}-72 a=t^{2}(t$ be a natural number). Then
$$
72 a=511^{2}-t^{2}=(511+t)(511-t) .
$$

Since $511+t$ and $511-t$ have the same parity, and $511+t \geqslant 511$, there are only the following possible cases:
(1) $\left\{\begin{array}{l}511+t=36 a \\ 511-t=2 .\end{array}\right.$
Adding the two equations gives $36 a+2=1022$, which has no integer solution;
(2) $\left\{\begin{array}{l}511+t=18 a, \\ 511-t=4 .\end{array}\right.$
Adding the two equations gives $18 a+4=1022$, which has no integer solution;
(3) $\left\{\begin{array}{l}511+t=12 a, \\ 511-t=6 .\end{array}\right.$
Adding the two equations gives $12 a+6=1022$, which has no integer solution;
(4) $\left\{\begin{array}{l}511+t=6 a \\ 511-t=12 .\end{array}\right.$
Adding the two equations gives $6 a+12=1022$, which has no integer solution;
(5) $\left\{\begin{array}{l}511+t=4 a, \\ 511-t=18 .\end{array}\right.$
Adding the two equations gives $4 a+18=1022$, solving for $a=251$.
(6) $\left\{\begin{array}{l}511+t=2 a \\ 511-t=36 .\end{array}\right.$
Adding the two equations gives $2 a+36=1022$, solving for $a=493$, but $493=17 \times 29$ is not a prime number, so it is discarded.
In summary, $a=251$.
At this point, the solution to Equation (3) is
$m=3$ or $m=\frac{502}{3}$ (discard).
Substituting $a=251, m=3$ into Equation (2) gives
$$
b=\frac{509 \times 3-6 \times 251}{3}=7 .
$$