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It is clear for $p=3$ ($x=0$, $y=1$ and $x=0$, $y=-1$) .

Let $x_0^2+3y_0^2=p$ , $\qquad $ $(x_0,y_0) \in \mathbb{Z^2}$ .

Reduction modulo 3 gives $$x_0^2 \equiv p \pmod 3$$

it follows that $p \equiv 1\pmod 3$ .

Now I can use the fact that the class number of $\mathbb{Q[\sqrt-3}]$ is 1 , but I do not know how to make use of it .

richrow
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AnabolicHorse
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  • For elementary approach one might use Thue's lemma for $a$ such that $a^2\equiv -3\pmod p$. Then, there is a nonzero pair $(x,y)$ such that $x,y<\sqrt{p}$ and $x\equiv ay\pmod p$. Now, it's not hard to prove that $x^2+3y^2=p$. For Thue's lemma see, for example, here https://en.m.wikipedia.org/wiki/Thue%27s_lemma – richrow May 16 '19 at 14:18
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    See https://math.stackexchange.com/questions/685958/let-p-be-prime-and-frac-3p-1-prove-that-p-is-of-the-form-p-a23b2/685982#685982 – Jack D'Aurizio May 16 '19 at 16:05
  • Also here – rtybase May 16 '19 at 17:26

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