Higher Order Diophantine Equation

Question

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.

Answer 1

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($X^4-8X^2+8,1000000$)"

           [ x  y  z ]
            
           [1, 1, 1]
           [7, 2, 31]
           [17, 45, 5311]
           [2273, 868, 305791]
           [50639, 98069, 23436127681]