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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.

  1. Every prime of the form $3k+1$ is expressible as $u^2 + 3v^2$ with $\gcd(u,v)=1$ in precisely one way.

  2. 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.

  3. 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.

  • 0
    In your first question, what is $a$ and $b$? You are only talking about a prime $p$ and some integers $u,v$2011-10-29
  • 1
    Have you tried anything yourself? If $p$ is odd and prime and not $3k+1$, what is it? and what would the implications be of such a prime dividing $a^2+3b^2$? in particular, what would it say about the quadratic character of $-3$? Have you seen the analogous theorems proved for $u^2+v^2$ and primes $p=4k+1$? If so, can you see how to tweak them to the current situation?2011-10-29
  • 0
    yes, I tried. But, I couldn't get.2011-10-29
  • 0
    See also this [related question.](http://math.stackexchange.com/questions/96101/is-every-mersenne-prime-of-the-form-x23-cdot-y2)2012-01-03

1 Answers 1