If an integer $n$ is such that $7n$ is the form $a^2 + 3b^2$, prove that $n$ is also of that form.

Question

I thought that looking at quad residues mod $7$ might??? help. But that didn't take me anywhere so apart from that I'm at a loss. I would appreciate any help. Thanks in advance.

score 4 · Accepted Answer · 2013-05-14T09:21:01.920

4

$\text{Make use of the identity }$$$\color{red}{(x^2+3y^2)(u^2+3v^2) = \underbrace{(xu\pm3yv)^2 + 3(xv\mp yu)^2}_{a^2 + 3b^2}}$$

$\text{Now we have}$ $$7 = 2^2 + 3 \times 1^2$$ $\text{i.e., }x=2 \text{ and }y=1$.

$\text{Use this to obtain $u$ and $v$.}$

$\textbf{EDIT}$:

$\text{Here is the full solution. The main claim is that if $7 \vert (a^2+3b^2)$, then}$

Either $7 \vert (a+2b)$ and $7 \vert (2a-3b)$
Or $7 \vert (a-2b)$ and $7 \vert (2a+3b)$

This can be checked by listing out the different possibilities for $a$ and $b$.

If $7 \vert (a+2b)$ and $7 \vert (2a-3b)$, then set $u = \dfrac{2a-3b}7$ and $v = \dfrac{a+2b}7$.

Else, if $7 \vert (a-2b)$ and $7 \vert (2a+3b)$, then set $u = \dfrac{3b-2a}7$ and $v = \dfrac{a+2b}7$.

This guarantess that $u$ and $v$ are integers. Hence, $\text{$n$ can be written as $u^2+3v^2$.}$

edited May 14 '13 at 09:21

answered May 14 '13 at 08:01

It isn't clear what you intend by this solution, especially given the fact that you've given new roles to the variables $a$ and $b$ from those in the problem statement. – Greg Martin May 14 '13 at 08:04
@GregMartin Changed the variable names. – May 14 '13 at 08:04
@GregMartin Have added the full solution. Hope it is clear now. – May 14 '13 at 08:16
@user17762, this is a special case of http://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity – lab bhattacharjee May 14 '13 at 08:22
@labbhattacharjee Yes. Thanks for the link. – May 14 '13 at 08:23
@user17762 If $a=5, b=1$, then $7 \mid a^2+3b^2$, but $7 \nmid 2b-a, 7 \nmid 2a+3b$. You need $u=\frac{2b \mp a}{7}, v=\frac{3b \pm 2a}{7}$. – Ivan Loh May 14 '13 at 08:58
@IvanLoh I have updated it. Essentially I need to regroup it in a different way. – May 14 '13 at 09:00
@user17762 Looks good now. (Ignore my now deleted comment if you saw it) – Ivan Loh May 14 '13 at 09:18

If an integer $n$ is such that $7n$ is the form $a^2 + 3b^2$, prove that $n$ is also of that form.

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