How do I prove that $x^2+8y^2$ does not properly represent 3?

Question

To me, it seems obvious that the binary quadratic form $x^2+8y^2$ does not properly represent 3. However, I have managed to prove that it does so I think I must be doing something stupid. I have used the following:

Let f be a a binary quadratic form and n an integer. We say that f properly represents n if there exists [x,y]∈$\mathbb Z^2$ such that (x,y)=1 and f(x,y)=n. (x,y) is defined as the greatest common divisor of x and y.

Lemma 5.3 (iii) Some form of discriminant d properly represents n if and only if $u^2\equiv d\pmod {4n}$

Then here is my 'proof':

Let $f(x,y)=x^2+8y^2$. Then, by Lemma 5.3(iii), $f(x,y)$ properly represents n=3 if and only if there is a solution to $u^2\equiv d\pmod{4\cdot3}$

$d=b^2-4ac=0^2-4\cdot1\cdot8=-32$ so

$u^2\equiv -32\pmod{12}$ which gives us

$u^2\equiv 4\pmod{12}$

which clearly has the solution $u=2$ so $f(x,y)$ properly represents n=3.

I know that this is wrong and it's very likely wrong for a stupid reason but I can't figure out what that is so any help would be appreciated.

Answer 1

Lemma $5.3$ is meant the following way:

Theorem: Let $n > 0$ and $d$ be given integers. There exists a binary quadratic form of discriminant $d$ that represents $n$ properly if and only if the congruence $x^2\equiv d\bmod 4d$ has a solution.

So we only know that there exists a binary quadratic form $f(x,y)$ with discriminant $d=-32$ representing $n=3$ properly. Certainly it is not $x^2+8y^2$.

Answer 2

As I said in a comment: one proof that $x^2 + 8 y^2$ does not represent the prime $3$ is that $3x^2 + 2xy + 3 y^2$ does represent $3,$ and these two forms are neither $SL_2 \mathbb Z$ equivalent nor are they "opposite." Put together, we could say they are $GL_2 \mathbb Z$ distinct.

You probably have access to tables for small negative discriminants; here is my own list. The principal genus has just the one form (class of) $x^2 + 8 y^2.$ The form that represents $3$ is in a different genus, $3 x^2 + 2 xy + 3 y^2.$ For example, regardless of choice of integers $x,y$ we never have $x^2 + 8 y^2 \equiv 3 \pmod 8.$

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./classGroup
Absolute value of discriminant? 
32
Impr 2 0 4
Discr  -32 = 2^5  class  number  2

 all  
      32:  < 1, 0, 8>    Square        32:  < 1, 0, 8>
      32:  < 3, 2, 3>    Square        32:  < 1, 0, 8>

 squares  
      32:  < 1, 0, 8>

 fourths  
      32:  < 1, 0, 8>


Discriminant        -32     h :    2     Squares :    1     Fourths :    1
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$