(-c^2+4ec-2e^2)X^4 + (-c^2+4ec-2e^2)Y^4 + (2c-2e)^2Z^4 = 2W^4

Existence of solution for diophantine equation ax^4 + by^4 + cz^4 + dw^4 = 0 are known if abcd is square number.
So, we are curious about whether above equation has a solution or not if abcd is not square number.
In particular, when does this equation have infinitely many integer solutions?

Parametric solutions of x^4 + ay^4 = z^4 + bt^4 are given below.
x^4 + ay^4 = z^4 + bt^4

We show diophantine equation (-c^2+4ec-2e^2)X^4 + (-c^2+4ec-2e^2)Y^4 + (2c-2e)^2Z^4 = 2W^4 has infinitely many integer solutions for p=9,19,129,243,289,801 with p<1000.

(-c^2+4ec-2e^2)X^4 + (-c^2+4ec-2e^2)Y^4 + (2c-2e)^2Z^4 = 2W^4...............................................................(1)

We use an identity p(t+1)^4+p(t)^4+r(t^2+at+b)^2=2(ct^2+dt+e)^2,............................................................(2)
with {a,b,d,p,r}={1, 1/2(c-2e)/(c-e), c, -c^2+4ec-2e^2, (2c-2e)^2}.

So, we look for the integer solutions {Z^2 = t^2+t+1/2(c-2e)/(c-e), W^2 = ct^2+ct+e}........................................(3)

By parameterizing the second equation and substituting the result to first equation, then we obtain quartic equation below.

u^2 =(-c^3+6c^2e-8e^2c+4e^3)k^4+(8e^2c-4c^2e)k^3+(-28c^2e+36e^2c+6c^3-8e^3)k^2+(8e^2c-4c^2e)k-c^3+6c^2e-8e^2c+4e^3..........(4)

This quartic equation has infinitely many rational solutions for |(p,r}|<100 below.

(c,e)= (7,4), (9,6), (9,7), (9,10), (9,13).

Hence we can obtain infinitely many integer solutions for equation (1) where (c,e)= (7,4), (9,6), (9,7), (9,10), (9,13).

Example:

(c,e)=(9,7): 73X^4 + 73Y^4 + Z^4 = 2W^4      : [X,Y,Z,W]=[887, 383, 298, 2199]

(c,e)=(9,10): 79X^4 + 79Y^4 + 4Z^4 = 2W^4    : [X,Y,Z,W]=[13, 3, 37, 47]