pX^4 + pY^4 = qZ^4 + qW^4

Existence of solution for diophantine equation ax^4 + by^4 + cz^4 + dw^4 = 0 are known if abcd is square number.
Furthermore, condition of a + b + c + d = 0 or having one known solution is necessary to have a rational solution.
Anyway, we show some parametric solutions of ax^4 + by^4 + cz^4 + dw^4 = 0 with abcd is square number.

General solutions of ax^4 + by^4 + cz^4 + dw^4 = 0 are given below.
ax^4 + by^4 + cz^4 + dw^4 = 0
ax^4 + by^4 + cz^4 + dw^4 = 0 Part 2

Parametric solutions of x^4 + hy^4 = z^4 + ht^4 are given below.
x^4 + hy^4 = z^4 + ht^4
Many numeric solutions of x^4 + hy^4 = z^4 + ht^4 are given below.
x^4 + hy^4 = z^4 + ht^4

We show diophantine equation pX^4 + pY^4 = qZ^4 + qW^4 has infinitely many integer solutions.
d,q are arbitrary.


pX^4 + pY^4 = qZ^4 + qW^4....................................................................(1)

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

So, we look for the integer solutions {Z^2 = t^2+(2-d)t-1/2d+1, W^2 = (-1+d)t^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 = (-1+d)k^4-4k^3+(2-2d)k^2+(12+8d^2-16d)k-9+17d+4d^3-12d^2...............................(4)

This quartic equation has infinitely many rational solutions for |(p,q}|<1000 below.

(p,q)= (17,2),(41,1),(113,1),(313,1),(257,2),(577,2).

Hence we can obtain infinitely many integer solutions for equation (1) where (p,q)= (17,2),(41,1),(113,1),(313,1),(257,2),(577,2).


Example:

(p,q)=(17,2): 17X^4 + 17Y^4 = 2Z^4 + 2W^4:  [X,Y,Z,W]=[31, 1, 49, 38],[14431, 15361, 12911, 30038],...etc.

(p,q)=(41,1): 41X^4 + 41Y^4 = Z^4 + W^4:    [X,Y,Z,W]=[1, 1, 1, 3],[11, 29, 61, 63],[5149, 17909, 38699, 37623],...etc.