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

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

We use an identity p(t+1)^4+p(t)^4=r(t^2+at+b)^2+s(ct^2+dt+e)^2,...........................(2)
with {a,b,e,p,q,r}={1, 0, 1/2d, 1/4sd^2, 1/4sd^2-scd+sc^2, 1/2sd^2-scd}.

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

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

u^2 =(4c-2d)k^4+2d.........................................................................(4)

Quartic curve (4) is transformed to the elliptic curve E.

x=(4c-2d)k^2, y=(4c-2d)uk.
E: y^2 = x^3+2(4c-2d)dx....................................................................(5)

If rank of E is greater than 0, then E has infinitely many rational solutions.
Hence we can obtain infinitely many integer solutions for equation (1) if rank of E is greater than 0.

Example:
|c|<5, d<10

[c, d] rank [ p, q, r, s] [X, Y, Z, W] 
[-4, 1] [1] [1, 81, 18, 4][9, 1, 3, 6]
[-4, 1] [1] [1, 81, 18, 4][9801, 2209, 4653, 5106]
[-4, 2] [1] [1, 25, 10, 1][9, 4, 6, 1]
[-4, 2] [1] [1, 25, 10, 1][25921, 144, 1932, 25919]
[-4, 5] [2] [25, 169, 130, 4][9, 1, 3, 14]
[-4, 5] [2] [25, 169, 130, 4][49, 9, 21, 74]
[-4, 5] [2] [25, 169, 130, 4][14161, 8649, 11067, 3886]
[-4, 8] [1] [16, 64, 64, 1][9, 4, 6, 14]
[-4, 8] [1] [16, 64, 64, 1][12769, 7056, 9492, 15934]
[-1, 2] [1] [1, 4, 4, 1][9, 4, 6, 7]
[-1, 2] [1] [1, 4, 4, 1][12769, 7056, 9492, 7967]
[-1, 8] [1] [16, 25, 40, 1][81, 16, 36, 158]
[0, 3] [1] [9, 9, 18, 4][49, 1, 7, 60]
[0, 7] [1] [49, 49, 98, 4][6241, 2209, 3713, 10920]
[1, 3] [1] [9, 1, 6, 4][9, 1, 3, 11]
[1, 3] [1] [9, 1, 6, 4][361, 625, 475, 13]
[1, 3] [1] [9, 1, 6, 4][201601, 27889, 74983, 246121]
[1, 4] [1] [4, 1, 4, 1][169, 1, 13, 239]
[1, 5] [1] [25, 9, 30, 4][121, 9, 33, 191]
[1, 5] [1] [25, 9, 30, 4][990025, 284089, 530335, 1526209]
[1, 6] [1] [9, 4, 12, 1][1089, 169, 429, 1871]
[1, 7] [1] [49, 25, 70, 4][529, 121, 253, 971]
[1, 8] [1] [16, 9, 24, 1][49, 16, 28, 94]
[1, 8] [1] [16, 9, 24, 1][3721, 1089, 2013, 7199]
[1, 9] [2] [81, 49, 126, 4][9, 1, 3, 19]
[1, 9] [2] [81, 49, 126, 4][25, 1, 5, 53]
[1, 9] [2] [81, 49, 126, 4][1521, 529, 897, 3071]
[1, 9] [2] [81, 49, 126, 4][3481, 2401, 2891, 5861]
[1, 9] [2] [81, 49, 126, 4][6241, 5329, 5767, 8711]
[1, 9] [2] [81, 49, 126, 4][42849, 47089, 44919, 22391]
[1, 9] [2] [81, 49, 126, 4][72361, 37249, 51917, 136771]
[4, 1] [1] [1, 49, -14, 4][2209, 1, 47, 1562]
[4, 2] [1] [1, 9, -6, 1][1, 4, 2, 7]
[4, 2] [1] [1, 9, -6, 1][121, 9, 33, 122]
[4, 2] [1] [1, 9, -6, 1][169, 225625, 6175, 390794]
[4, 2] [1] [1, 9, -6, 1][2209, 784, 1316, 2593]
[4, 3] [1] [9, 25, -30, 4][1521, 529, 897, 2042]
[4, 5] [1] [25, 9, -30, 4][529, 1521, 897, 2042]
[4, 6] [1] [9, 1, -6, 1][4, 1, 2, 7]
[4, 6] [1] [9, 1, -6, 1][9, 121, 33, 122]
[4, 6] [1] [9, 1, -6, 1][225625, 169, 6175, 390794]
[4, 6] [1] [9, 1, -6, 1][784, 2209, 1316, 2593]
[4, 7] [1] [49, 1, -14, 4][1, 2209, 47, 1562]
[4, 9] [1] [81, 1, 18, 4][1089, 2209, 1551, 1702]