Solving simultaneous (non linear) integer equations (a bit like conics) I'm looking for all solutions, (x,y,s,t) in the integers, for the two simultaneous equations...
$$
7x^2 - y^2 = 3s^2\\
7y^2 - x^2 = 3t^2
$$
I have two solutions $(x,y,s,t) = (2,1,3,1)$ and $(751,422,1121,477)$.
I'm also interested in solving the more general cases...
$$
Ax^2 + By^2 = Cs^2\\
Ay^2 + Bx^2 = Ct^2
$$
where $A + B = 2C$
Is there a heading I can search under for more info?
Thanks.
 A: There are an infinite number of solutions, but they get large quite quickly. Solving this system is a standard application of elliptic curves.
The quadric $7x^2-y^2=3s^2$ has the simple solution $x=1, y=2, s=1$, which allows us to find the parametric solution $x=3k^2-6k+7, y=2(3k^2-7)$. 
Substituting into $7y^2-x^2=3t^2$ gives the quartic
\begin{equation*}
t^2=81k^4+12k^3-418k^2+28k+441
\end{equation*}
which has an obvious rational point when $k=0$, and so is birationally equivalent to an elliptic curve.
Standard methods find this curve to be
\begin{equation*}
v^2=u^3-97u^2+2352u
\end{equation*}
with
\begin{equation*}
k=\frac{6v-u}{3(9u-448)}
\end{equation*}
The elliptic curve has $7$ finite torsion points at $(0,0)$, $(48,0)$, $(49,0)$, $(42,\pm 42)$ and $(56, \pm 56)$. It also has rank $1$ with generator $(21,126)$.
This generator gives $k=-35/37$, and hence the second solution quoted.
Doubling the generator gives the following solution
\begin{equation*}
x=124344271, \, y=56190422, \, s=187147999, \,t=47046243
\end{equation*}
Further computation can give more solutions.
Allan Macleod
A: Above equation shown below:
$Ax^2 + By^2 = Cs^2\\
Ay^2 + Bx^2 = Ct^2$
"OP" want's parametric solution for variable's $(x,y,s,t)$. Allen Macleod has kindly provided solution only for variable's $(x,y)$. By extension, the parametric solution given by Allen Macleod for variables $(s,t)$ would be:
$s=(p-q)(p^8+16p^5q^3+14p^4q^4+16p^3q^5+q^8)$
$t=(p+q)(p^8-16p^5q^3+14p^4q^4-16p^3q^5+q^8)$
For $(p,q)=(2,1)$ we get back solution given by "OP" as
$(A,B,C)=(7,-1,3)$
$(x,y,s,t)=(751,422,1121,477)$
A: With regard to the general problem
\begin{equation*}
2Ax^2+2By^2=(A+B)s^2 \hspace{2cm} 2Ay^2+2Bx^2=(A+B)t^2
\end{equation*}
solutions do NOT exist for any combination of $A$ and $B$.
For example, for $A=5, B=-1$, there are no solutions since the quadric $5x^2-y^2=2s^2$ is not locally soluble at the prime $p=5$. 
Allan Macleod
A: The general problem
\begin{equation*}
Ax^2+By^2=Cs^2 \hspace{2cm} Ay^2+Bx^2=Ct^2
\end{equation*}
with $A=Q+P, B=Q-P$ and $Q=p^2-q^2, P=2pq$ leads to an elliptic curve with rank at least $1$ for any $(p,q)$ with $|p| \ne |q|$ and $pq \ne 0$. This gives an infinite number of possible parametric solutions.
One such is
\begin{equation*}
x=q(3p^8-4p^6q^2+14p^4q^4+4p^2q^6-q^8)
\end{equation*}
\begin{equation*}
y=p(p^8-4p^6q^2-14p^4q^4+4p^2q^6-3q^8)
\end{equation*}
The derivation is a standard (if boring) computation, using a symbolic algebra package.
A: Above equation shown below:
$Ax^2 + By^2 = Cs^2\\
Ay^2 + Bx^2 = Ct^2$
Above has solution:
$(x,y,s,t)= (36,31,41,24)$
$(A,B,C)=(144,-77,67)$
For detail's see the below mentioned link:
Non-trivial solutions of $Ax^2+By^2=Cs^2$ and $Ay^2+Bx^2=Ct^2$, where $A=p^2-q^2+2pq$, $B=p^2-q^2-2pq$, $C=p^2-q^2$ for integer $p$ and $q$
