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.1.IntroductionSimultaneous 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.2.Theorem(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}.3.Example

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