Integers $d$ for which the negative Pell equation is soluble for both $d$ and $2d$?

Question

Let $\text{NPE}_d$ denote the negative Pell equation: $$ x^2-dy^2=-1$$ Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y.

we know that (in this paper archive):

Theorem : The equation $\text{NPE}_d$ has integer solutions if and only if there exist two integers $a(d)=a$ and $b(d)=b$ such that $d=a^2+b^2$ and there exists a Pythagorean triplet $(A,B,C)$ such that $|aA-bB|=1$ and in this case $(Ab+Ba,C)$ is a solution.

Obviously if $\text{NPE}_d$ has integer solutions then $d$ cannot be divisible by any prime $p$ such that $p=3\mod 4$.

My question: Is there any characterization for the integers $d$ for which $\text{NPE}_d$ and $\text{NPE}_{2d}$ have both integer solutions.

I used the characterization above, but I can't link the couple $(a(d),b(d))$ to $(a(2d),b(2d))$ because the theorem doesn't give us much information

The sequence of the elements $d$ for which $\text{NPE}_d$ is soluble is OEIS A031396.

I posted this question in math.exchange and does not receive any answer

Answer 1

I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:

It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).

Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ (the latter with $2^{(p-1)/4}\equiv-1\bmod p$), the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)

So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$, or if $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, both $\text{NPE}_d $ and $ \text{NPE}_{2d} $ have integer solutions.

Update

I noticed that if we represent $ 2p = a^2 + b^2 $, if it holds that $a\equiv\pm 3\bmod 8$ and $ b\equiv\pm 3\bmod 8 $, then $ x^2 - 2py^2 = -1 $ is not solvable, but I don't know how to prove it. This maybe is related with the GLW theorem that the OP highlithed

Answer 2

(Too long for a comment.)

The sequence of integers such that $x^2-py^2=-1$ is solvable is given by,

$$p = 1, 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101,\dots$$

which is A031396, while that of $x^2-2q y^2 = -1$ is (if I did my code right),

$$q = 1, 5, 13, 25, 29, 37, 41, 53, 61, 65, 85, 101, \dots$$

and is not yet in the OEIS.

Point 1: The sequence $p$ does not contain non-square $q$ as a subset.

With limited data, it seems to be the case. But the first missing nonsquare value is $q=221$, since $x^2-221y^2=-1$ is not solvable, while $x^2-2\cdot221y^2=-1$ is.

Point 2: There is an infinite number of intersections between $p$ and $q$.

Proof: We use the identities,

$$m^2-(m^2+1)\cdot 1^2 = -1$$

$$(2n+1)^2-2\cdot(2n^2+2n+1)\cdot 1^2 = -1$$

Equate,

$$m^2+1 = 2n^2+2n+1$$

and turns out to be a well-known Pell equation in disguise,

$$(2n+1)^2-2m^2 = 1$$

$$u^2-2v^2=1$$

with solutions $(u,v) = (3,2),\,(17,12),\,(99,70),\dots$ and proves there is an infinite number of $d$ such that

$$x^2-dy^2 = -1$$

$$x^2-2dy^2 = -1$$

is both solvable.

P.S. However, to characterize all $d$ seems to be difficult.