2mnX^4 + (m^2-n^2)Y^4 = 2mnZ^4 + (m^2-n^2)W^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 2mnX^4 + (m^2-n^2)Y^4 = 2mnZ^4 + (m^2-n^2)W^4 has infinitely many integer solutions.
m,n are arbitrary.

2mnX^4 + (m^2-n^2)Y^4 = 2mnZ^4 + (m^2-n^2)W^4..........................(1)

We consider below equation.

p(at+s)^4 + q(bt-1)^4 = p(at-s)^2 + q(bt+1)^2.

(8pa^3s-8qb^3)t^3+(8pas^3-8qb)t=0......................................(2)

To obtain a rational solution of (2) for t, we have to find the rational solution of (3).

v^2 = -a^4s^4p^2+(qb^3as^3+a^3sqb)p-q^2b^4.............................(3)

Let consider equation (3) is a quadratic equation for p.

Substitute p=b/s, q=a to equation (3), then v^2 = -a^2b^2(s-1)(s+1)(a-b)(a+b).

Let b=m^2+n^2, a=m^2-n^2, s = 1/2(m^2+n^2)/(mn), then we obtain p = 2mn, q = m^2-n^2, t = 1/4(-m^2+n^2)/(m^2n^2).

Thus we obtain a parametric solution.

(p,q)=(2mn, m^2-n^2).

X = m^4-2m^2n^2+n^4-2m^3n-2mn^3
Y = m^4-n^4+4m^2n^2
Z = m^4-2m^2n^2+n^4+2m^3n+2mn^3
W = m^4-n^4-4m^2n^2

m,n are arbitrary.

Furthermore, we can find other parametric solution by using this solution.
See ax^4 + by^4 + cz^4 + dw^4 = 0

Hence by repeating this process, we can obtain infinitely many integer solutions for equation (1).


Example:

(m,n)=(2,1): 4X^4+3Y^4 = 4Z^4+3W^4
             (X,Y,Z,W)=(11, 31, 29, 1),(264009744251, 722953698209, 676722505229, 70657231681).

(m,n)=(3,2): 12X^4+5Y^4 = 12Z^4+5W^4

(m,n)=(4,1): 8X^4+15Y^4 = 8Z^4+15W^4