First of all, I am very thankful to this site. I just came to know this site by google. I have seen some number theory question on this site. The discussion between learner and author is quite good and interesting. I would like to know the proof of following questions. If any one answered, I am very grateful of them.
Every prime of the form $3k+1$ is expressible as $u^2 + 3v^2$ with $\gcd(u,v)=1$ in precisely one way.
The general primitive solution in integers of the equation $x^2 + 3y^2 = N^3$ for odd $N$ is given by $x = u(u^2 - 9v^2)$ and $y = 3v(u^2 - v^2)$ where $u$ and $v$ are co-prime integers.
If an integer is representable in the form $a^2 + 3b^2$ with $\gcd(a,3b)=1$, then its only odd prime factors are of the form $p = 3k+1$.
Once again thanks for all team members of this site.