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Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the diophantine equation,

$$x^2+dy^2 = 2^{2+h(-d)}\tag{1}$$

with odd integers $x,y$?

For example, one can indeed solve,

$$x^2+7y^2 = 2^3$$

$$x^2+23y^2 = 2^5$$

$$x^2+47y^2 = 2^7$$

$$x^2+71y^2 = 2^9$$

which have $d$ with $h(-d) = 1,3,5,7$, respectively, and it is solvable for all such $d$ with $h(-d) \leq 25$ in the link given above (I tested them all), but it would be nice to know if it is true in general.

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  • $\begingroup$ note that your $d$ must be prime. $\endgroup$
    – Will Jagy
    Nov 11, 2013 at 22:36
  • $\begingroup$ @WillJagy: Yes. $\endgroup$ Nov 11, 2013 at 22:39
  • $\begingroup$ Don't you mean $-d=1\pmod{8}$ ? $\endgroup$
    – Aurel
    Nov 11, 2013 at 22:46
  • $\begingroup$ Oops, typo. I'll edit it. $\endgroup$ Nov 11, 2013 at 22:48

2 Answers 2

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Because $d=-1\pmod{8}$, $2$ splits in $K=\mathbb{Q}(\sqrt{-d})$. Let $\mathfrak{p}$ be a prime above $2$. Then $\mathfrak{p}^{h(-d)}$ is principal and has norm $2^{h(-d)}$. Let $z=x+\frac{1+\sqrt{-d}}{2}y$ be a generator, so $z$ has norm $2^{h(-d)}$. But this can be rewritten as $(2x+y)^2+dy^2 = 4\cdot 2^{h(-d)}$. Since $d$ is odd, $2x+y$ and $y$ are both odd unless $2$ divides $z$. But if it were so, $(2)$ would divide $\mathfrak{p}^{h(-d)}$ so $\bar{\mathfrak{p}}$ would divide $\mathfrak{p}^{h(-d)}$, which it does not (uniqueness of factorization).

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  • $\begingroup$ Nice. I think this should read $(2x+y)^2+dy^2 = 4\cdot 2^{h(-d)}$. But then, why should $x$ be odd, as asked in the question? $\endgroup$
    – abx
    Nov 12, 2013 at 11:35
  • $\begingroup$ @abx Indeed it is $2x+y$ and not $x+y$, thanks! The notation is not the same as in the question, so here we ask that $2x+y$ and $y$ are odd. If $y=2y'$ was even, then $4(x+y')^2+4d(y')^2=0\pmod{8}$ so $x$ would be even too. So $x$ and $y$ would be both even and $z$ would be divisible by $2$. $\endgroup$
    – Aurel
    Nov 12, 2013 at 13:00
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Or, put the cheesy way: with $d = p \equiv 7 \pmod 8,$ there is a primitive positive quadratic form with coefficients $$ \langle 2, 1 , \frac{p+1}{8} \rangle. $$ As a result, with class number $h,$ by repeated Gauss composition we know the principal form $$ \langle 1, 1 , \frac{p+1}{4} \rangle $$ integrally represents $2^h.$ Note that, from Theorem 4.12 on page 64 of Buell, the composition algorithm of Shanks, as we continue with repeated Gauss composition, the middle coefficient remains $\equiv 1 \pmod 4,$ which means that we are showing that the principal form primitively represents $2^h.$

Next, we have the same class number in the genus of $$\langle 1, 0 , p \rangle, $$ although there is no longer a primitive form that represents $2$ itself. Page 118, Theorem 7.5 (a) in Buell, Binary Quadratic Forms.

Finally, $$ 4 (x^2 + xy +\frac{p+1}{4} y^2 ) = 4 x^2 + 4 x y + (p+1) y^2 = (2x+y)^2 + p y^2, $$ so we can represent $4 \cdot 2^h$ as $u^2 + p v^2.$

EDIT, Tuesday: Note that $x^2 + xy +\frac{p+1}{4} y^2$ and $u^2 + p v^2$ represent exactly the same odd numbers, as $\frac{p+1}{4}$ is even, so to get an odd number we have $x^2 + xy = x(x+y)$ odd, so $x$ is odd and $y$ is even; take $u=x, v = y/2.$

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