1.Introduction

We show that simultaneous equation {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} has infinitely many integer solutions.
                 
                 
2.Theorem
     
Simultaneous diophantine equation {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} has infinitely many integer solutions.


 
Proof.

a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4................................................(1)

Let a = p(x+s), b = q(x-s), c = r(x+t), d =x-t, e = p(x-s), f = q(x+s), g = r(x-t), h =x+t then equation (1) becomes to below equation.

(8p^4s-8q^4s+8r^4t-8t)x^2+(-8t^3+8p^4s^3-8q^4s^3+8r^4t^3)=0......................(2)

Since this is a quadratic equation in x, for x to be rational number, the discriminant of the equation must be square number.

Let U=s/t then we obtain

V^2 = -(r^4-1)^2t^4-(p^4s-q^4s)(r^4-1)t^3-(r^4-1)(p^4s^3-q^4s^3)t-(p^4s-q^4s)(p^4s^3-q^4s^3).

Let (p,q,r)=(2,3,4), then we obtain V^2 = -65025U^4+16575U^3+16575U-4225.........(3)

Quartic equation (3) has a rational point Q(U,V)=(13/51,0), then this quartic equation is birationally equivalent to an elliptic curve below.

Y^2+YX = X^3-X^2-82419 -19004491.................................................(4)
Transformation is given, 

U = (13X+8450)/(51X+2142), V = 1/51(79040Y+39520X)/((X+42)^2)
X = (8450-2142U)/(51U-13), Y = (201552V-1146990U+274625+273105U^2)/(13005U^2-6630U+845).

Elliptic curve (4) has rank 1 and generator P(X,Y)=(1934 , 83013).

This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points.

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

Similarly, we can say that there are infinitely many integer solutions for (p,q,r)=(1,2,4),(2,3,2),(2,3,3), etc. 

Q.E.D.


3.Example

(p,q,r)=(2,3,4).

V^2 = -65025U^4+16575U^3+16575U-4225.

We obtain rational point 2Q(U,V)=(1733/3291, 158338400/3610227).
Hence t=1733, s=3291, x = 10417/5.

{a,b,c,d}={26872, 9057, 38164, 876}.
{e,f,g,h}={6038, 40308, 3504, 9541}.