pX^4 + pY^4 = 2Z^4 + W^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 + W^4 has infinitely many integer solutions for p= 1+2b^4.
b is arbitrary.
 
pX^4 + pY^4 = 2Z^4 + W^4..........................................................................(1)

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

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

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

u^2 =k^4+(-4b+4b^3)k^3+(-12b^4+2b^2)k^2+(4b^3+4b+16b^5)k-4b^4-4b^2-8b^6-1.........................(4)

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

Y^2+(-4b+4b^3)YX+(8b^3+8b+32b^5)Y = X^3+(-4b^4-2b^2-4b^6)X^2+(16b^4+16b^2+32b^6+4)X-192b^8-112b^6-192b^10-48b^4-8b^2-128b^12

The corresponding point is P(X,Y)=( 4b^4+2b^2+4b^6, -24b^5-16b^9-8b ).

Hence we get 2P(X,Y)=( 1/4(4b^8+16b^6-12b^4+1)/(b^2), -1/8(52b^8+80b^6+18b^4+24b^12+48b^10-4b^2-1)/(b^3) ).

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 = 1+2b^4.
b is arbitrary.

m=2:
X = 4b^8+20b^4+1
Y = -8(2b^4-1)b^2
Z = b(-3-20b^4+4b^8)
W = -1+20b^4+12b^8

m=3:
X = (4b^8+20b^4+1)(16b^16+416b^12+152b^8+104b^4+1)
Y = -8(2b^4-1)(4b^8-20b^4-3)b^2(12b^8+20b^4-1)
Z = (64b^24-3008b^20-6096b^16-9120b^12-932b^8+228b^4+5)b
W = 1-188b^4-1524b^8-9120b^12-3728b^16+3648b^20+320b^24
.
.
etc.

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

Example:

b=1: 3X^4+3Y^4 = 2Z^4+W^4    : see details 3X^4+3Y^4 = 2Z^4+W^4

b=2: 33X^4+33Y^4 = 2Z^4+W^4  : see details 33X^4+33Y^4 = 2Z^4+W^4