<s>[INST] System of quadratic homogeneous Diophantine equations Is there a method for determining if a system of quadratic diophantine equations has any solutions?
My specific  question is:
\begin{align*}
a^2+b^2&=c^2\\[4pt]
2a^2+b^2&=d^2\\
\end{align*}
I want to know if there are any positive integers $a, b, c, d$ which satisfy both equations.
The question came from my student and I cannot answer her in class.
 [/INST] Suppose $(a,b,c,d)$ is a solution to the system
$$
\left\lbrace
\begin{align*}
&a^2+b^2=c^2\\[4pt]
&2a^2+b^2=d^2\\[4pt]
\end{align*}
\right.
$$
where $a,b,c,d$ are positive integers, and where $d$ is as small as possible.

If $a,b$ have a common factor $g > 1$, then $c$ and $d$ must also be divisible by $g$, hence 
$$\left(\frac{a}{g},\frac{b}{g},\frac{c}{g},\frac{d}{g}\right)$$
would also be a solution, contrary to the assumed minimality of $d$.

Hence $\gcd(a,b)=1$.

Suppose $b$ is even.

Then from $2a^2+b^2=d^2$, it follows that $d$ is even. 

But then from $2a^2=d^2-b^2$, it follows that $2a^2$ is a multiple of $4$, hence $a$ is even, contrary to $\gcd(a,b)=1$.

Hence $b$ is odd.

Since $\gcd(a,b)=1$ and $b$ is odd, using the standard form for primitive Pythagorean triples, it follows that the equation 
$$a^2+b^2=c^2$$
holds if and only if
$$
\left\lbrace
\begin{align*}
a&=2st\\[4pt]
b&=s^2-t^2\\[4pt]
c&=s^2+t^2\\[4pt]
\end{align*}
\right.
$$
for some positive integers $s,t$ such that

*

*$s > t$.$\\[4pt]$

*$\gcd(s,t)=1$.$\\[4pt]$

*One of $s,t$ is even (and the other is odd).


Then the equation $2a^2+b^2=d^2$ can be recast as
$$
2(2st)^2+(s^2-t^2)^2=d^2
$$
or equivalently
$$
s^4+6s^2t^2+t^4=d^2
$$
hence, letting $x={\Large{\frac{t}{s}}}$ and $y={\Large{\frac{d}{s}}}$, it follows that the equation
$$y^2=x^4+6x^2+1$$
has a rational point $(x,y)$ with $0 < x < 1$.

That's as far as I can go.

I suspect there is no such rational point, but if so, I doubt that it can be proved in an elementary way.

Using more advanced methods, perhaps there is a way to convert the equation $y^2=x^4+6x^2+1$ to the equation of an elliptic curve in such a way that the non-existence of qualifying rational points on the curve $y^2=x^4+6x^2+1$ can be inferred from the non-existence of qualifying rational points on the elliptic curve.
</s>