I am trying to obtain the general solution of the following Diophantine equation.
$$x^4-8x^2y^2+8y^4=z^2$$
My approach was to rewrite the equation.
\begin{align}
\begin{split}
& x^4+z^2=2(2y^2-x^2)^2\\
\implies & p^2+q^2=2r^2
\end{split}
\end{align}
Let $a^2+b^2=r^2$. After some manipulation,
\begin{align}
\begin{split}
& (a^2+b^2)+(a^2+b^2)=2r^2 \\
& (a^2+b^2+2ab)+(a^2+b^2-2ab)=2r^2 \\
& (a+b)^2+(a-b)^2=2r^2
\end{split}
\end{align}
$(a,b,c)$ forms a Pythagorean triple. So we get
\begin{align}
\begin{split}
&(a,b,c)=(k(m^2-n^2),2kmn,k(m^2+n^2))\\
&(p,q,r)=(x^2,y^2,z)=(k(m^2-n^2+2mn),k(m^2+mn),k(m^2+n^2))
\end{split}
\end{align}
I am stuck here. Any help will be appreciated.
Higher Order Diophantine Equation
diophantine equationsnumber theorypythagorean triples
Best Answer
Althogh this method is different from your method, the solution is easy to derive.
Let $U=\frac{\large{x}}{\large{y}}, V=\frac{\large{z}}{\large{y^2}}$ then we get $V^2 = U^4-8U^2+8$.
This quartic equation is birationally equivalent to the elliptic curve below.
$Y^2 = X^3 + X^2 -3X + 1$
$U = \frac{Y}{(X-1)}, V = \frac{X^3-3X^2+X+1}{(X-1)^2}$
$X = \frac{1}{2}V+\frac{1}{2}U^2-1, Y = \frac{1}{2}U(V+U^2-4)$
This elliptic curve has rank $1$ with generator $(0,-1)$.
The rank and generator are obtainded using mwrank(Cremona).
Hence this curve has infinitely many rational points.
Thus, we get infinitely many integer solutions.
Example:
First, we get a rational solution $(X,Y)=(\frac{-56}{25}, \frac{153}{125})$ using mwrank or Magma.
From $U = \frac{Y}{(X-1)}, V = \frac{X^3-3X^2+X+1}{(X-1)^2}$, we get $(U,V)=(\frac{-17}{45}, \frac{-5311}{2025})$
Hence we get $(x,y,z)=(17,45,5311).$
More solutions using group law:
Let $P(X,Y)=(0,-1)$, then we get
$6P(X,Y)=(\frac{30547621}{2340900}, \frac{-173783774969}{3581577000})$
$7P(X,Y)=(\frac{-6037269840}{7661325841}, \frac{-1253783210541679}{670588189536889}).$
Hence we get the corresponding solutions
$(x,y,z)=(32721479, 8125830, 784659483951841)$
$(x,y,z)=(10712341919, 10244481689, 70259696347563139199).$
Some solutions with $x<1000000$ using PARI-GP as follows.
"hyperellratpoints($X^4-8X^2+8,1000000$)"