pX^4 + pY^4 + 2Z^4 = sW^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 pX^4 + pY^4 + 2Z^4 = sW^4 has infinitely many integer solutions.
p = -2+a^2, s = a^2.

pX^4 + pY^4 + 2Z^4 = sW^4...........................................................................(1)

We use an identity p(at+a)^4+p(t)^4+r(t^2+a^2t+a^2)^2=s(ct^2+2*ct+a^2)^2,..........................(2)
with s = p+r, r = 2p/(a^2-2), c = a^2-1.

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

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

u^2 = k^4+(4a^2-6)k^2+(-4a^4-8+12a^2)k-3+8a^2-5a^4+a^6.............................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.

Y^2+(-8a^4-16+24a^2)Y = X^3+(4a^2-6)X^2+(12-32a^2+20a^4-4a^6)X+240a^2-248a^4+104a^6-16a^8-72.

The corresponding point is P(X,Y)=(-4a^2+6, 16+8a^4-24a^2  ).

Hence we get 2P(X,Y)=( 1/4a^4+a^2+3, 7/4a^4-33/2a^2+9+1/8a^6 ).

This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points.

This quartic equation has infinitely many parametric solutions below.
p = -2+a^2, s = a^2.

m=2:
X = 8a(a^2+2)
Y = (a^2-4a-2)(a^2+4a-2)
Z = -4-20a^2+3a^4
W = a^4+20a^2-12

m=3:
X = 8a(a^4+20a^2-12)^2(a^2+2)(-4-20a^2+3a^4)
Y = (a^4+20a^2-12)(a^2-4a-2)(a^2+4a-2)(a^4-8a^3-20a^2-16a+4)(a^4+8a^3-20a^2+16a+4)
Z = (a^4+20a^2-12)(64+3008a^2-6096a^4+9120a^6-932a^8-228a^10+5a^12)
W = (a^4+20a^2-12)(a^12+188a^10-1524a^8+9120a^6-3728a^4-3648a^2+320)
.
.
etc.

Hence we can obtain infinitely many integer solutions for equation (1).

Example:

a=4: 7X^4 + 7Y^4 + Z^4 = 8W^4      : see details 7X^4 + 7Y^4 + Z^4 = 8W^4

a=5: 23X^4 + 23Y^4 + 2Z^4 = 25W^4  : see details 23X^4 + 23Y^4 + 2Z^4 = 25W^4

a=7: 47X^4 + 47Y^4 + 2Z^4 = 49W^4  : see details 47X^4 + 47Y^4 + 2Z^4 = 49W^4