pr^2X^4 + 4p(p+r)^2Y^4 + r^3Z^4 = r^2(p+r)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 pr^2X^4 + 4p(p+r)^2Y^4 + r^3Z^4 = r^2(p+r)W^4 has infinitely many integer solutions.
p,r are arbitrary.

pr^2X^4 + 4p(p+r)^2Y^4 + r^3Z^4 = r^2(p+r)W^4.............................................(1)

We use an identity p(t+1)^4+q(t)^4+r(t^2+at+1)^2=(p+r)(ct^2+dt+1)^2,......................(2)
with (a,c,d,q)=(-2p/r, (2p+r)/r, 0, 4p(r^2+p^2+2pr)/(r^2)).

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

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

u^2 = r^4k^4+r(8pr^2+2r^3)k^2+r(8pr^2+16p^2r)k+r(4p^2r+8p^3+r^3)..........................(4)

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

Y^2+(16r^5p+32r^4p^2)Y = X^3+(8r^3p+2r^4)X^2+(-16r^6p^2-32r^5p^3-4r^8)X-192r^9p^3-32r^10p^2-256r^8p^4-32r^11p-8r^12

The corresponding point is P(X,Y)=( -8pr^3-2r^4, -16r^5p-32r^4p^2 ).

Hence we get 2P(X,Y)=( 6r^4+4pr^3+p^2r^2, 16r^5p-10r^4p^2+16r^6-p^3r^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.

m=2:
X = -8r^2-8pr+p^2
Y = 4(2r+p)r
Z = 3p^2-4pr-8r^2
W = p^2+12pr+8r^2

m=3:
X = (p^2+12pr+8r^2)(-8r^2-8pr+p^2)(-64r^4-128r^3p-112p^2r^2-48p^3r+p^4)
Y = 4(2r+p)(3p^2-4pr-8r^2)r(p^2+12pr+8r^2)^2
Z = (p^2+12pr+8r^2)(5p^6-84p^5r-728r^2p^4-832r^3p^3+576r^4p^2+1280r^5p+512r^6)
W = (p^2+12pr+8r^2)(p^6+100p^5r+104r^2p^4+576r^3p^3+1856r^4p^2+1792r^5p+512r^6)
.
.
etc.

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

Example:

(p,r)=(1,2): X^4 + 9Y^4 + 2Z^4 = 3W^4      : see details X^4 + 9Y^4 + 2Z^4 = 3W^4

(p,r)=(7,1): 7X^4 + 7Y^4 + Z^4 = 8W^4      : see details 7X^4 + 7Y^4 + Z^4 = 8W^4