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 parametric solutions.
We use old result {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} below.
    {a^4 + b^4 + c^4 + d^4 = e^4 + f^4 + g^4 + h^4, abcd=efgh} has a follwing parametric solution.

    a = p(m + r^4 - q^4)          e = p(m - r^4 + q^4)
    b = q(m + p^4 - w^4)          f =-q(-m + p^4 - w^4)
    c =-r(-m + p^4 - w^4)         g = r(m + p^4 - w^4)
    d = w(m - r^4 + q^4)          h = w(m + r^4 - q^4)
    
    Condition: p^4 + q^ 4 = r^4 + w^4

First, we show diophantine equation p^4+q^4=r^4+w^4 has infinitely many parametric solutions.

2.Lemma

Diophantine equation p^4+q^4=r^4+w^4 has infinitely many parametric solutions.


Proof.

p^4+q^4=r^4+w^4.......................................................................................(1)

Let p=kt+u, q=nt+v, r=kt-v, w=nt+u, then equation (1) becomes to below equation.

(4vk^3+4vn^3-4un^3+4uk^3)t^2+(-6u^2n^2-6v^2k^2+6v^2n^2+6u^2k^2)t-4u^3n+4u^3k+4v^3k+4v^3n=0............(2)

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

Let U=k/n then we obtain

V^2 = (-28u^4-28v^4-64vu^3-72v^2u^2-64v^3u)U^4
    +(-64v^4+64u^4+64vu^3-64v^3u)U^3
    +(-72u^4+144v^2u^2-72v^4)U^2
    +(-64vu^3+64v^3u-64v^4+64u^4)U
    +64vu^3-72v^2u^2+64v^3u-28u^4-28v^4...............................................................(3)
    
Quartic equation (3) has a rational point Q(U,V)=( (u^3-v^3)/(u^3+v^3), 24u^3v^3(u-v)/((u^2-uv+v^2)^2(u+v)) ), then this quartic equation is birationally equivalent to an elliptic curve below.

Y^2-8/3(2v^8-2v^7u+7v^6u^2-5v^5u^3-8u^4v^4-5u^5v^3+7u^6v^2-2u^7v+2u^8)*Y*X/(v^2u^2(u^2-uv+v^2))-2304v^3u^3(v^6+uv^5-u^2v^4-2u^3v^3-u^4v^2+u^5v+u^6)*Y/((u^2-uv+v^2)^3)
= X^3-32/9(2v^16-4v^15u+16v^14u^2-24u^3v^13+32u^4v^12-2u^5v^11-6u^6v^10+41u^7v^9-102u^8v^8+41u^9v^7-6u^10v^6-2u^11v^5+32u^12v^4-24u^13v^3+16u^14v^2-4u^15v+2u^16)*X^2/(u^4v^4(u^2-uv+v^2)^2)
+9216(7v^4-12v^3u+10v^2u^2-12vu^3+7u^4)u^6v^6*X/((u^2-uv+v^2)^4)
-32768u^2v^2(14v^20-52uv^19+180u^2v^18-424v^17u^3+734u^4v^16-858u^5v^15+702v^14u^6-213u^7v^13-1018u^8v^12+1979v^11u^9-2088u^10v^10+1979u^11v^9-1018v^8u^12-213u^13v^7+702u^14v^6-858v^5u^15+734u^16v^4-424u^17v^3+180v^2u^18-52u^19v+14u^20)/((u^2-uv+v^2)^6)

The point corresponding to point Q is P(X,Y)=( 32/9(2v^16-4v^15u+16v^14u^2-24u^3v^13+32u^4v^12-2u^5v^11-6u^6v^10+41u^7v^9-102u^8v^8+41u^9v^7-6u^10v^6-2u^11v^5+32u^12v^4-24u^13v^3+16u^14v^2-4u^15v+2u^16)/(u^4v^4(u^2-uv+v^2)^2),
512/27(2v^18+15v^16u^2+12u^4v^14-43u^6v^12+6u^8v^10+6u^10v^8-43u^12v^6+12u^14v^4+15u^16v^2+2u^18)/(v^6u^6) ).

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

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


Q.E.D.



3.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 parametric solutions.

 
Proof.

According to old result, simultaneous diophantine equation {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh}  has a follwing parametric solution.

a = p(m + r^4 - q^4),          e = p(m - r^4 + q^4)
b = q(m + p^4 - w^4),          f =-q(-m + p^4 - w^4)
c =-r(-m + p^4 - w^4),         g = r(m + p^4 - w^4)
d = w(m - r^4 + q^4),          h = w(m + r^4 - q^4)
    
Condition: p^4 + q^ 4 = r^4 + w^4.
m is arbitrary.

From Lemma, diophantine equation p^4+q^4=r^4+w^4 has infinitely many parametric solutions, then
simultaneous equation {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} has infinitely many parametric solutions.


Q.E.D.


4.Example

Case of Q(U,V)=( (u^3-v^3)/(u^3+v^3), 24u^3v^3(u-v)/((u^2-uv+v^2)^2(u+v)) ).
To simplify the result, let v=1.

p = u^7+u^5-2u^3-3u^2+u
q = u^6+3u^5+u^2-2u^4+1
r = -u^6+3u^5-u^2+2u^4-1
w = u^7+u^5-2u^3+3u^2+u

a = u(u^4-3u+u^6-2u^2+1)(-1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)
b = (3u^5+u^2+u^6-2u^4+1)(-1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)
c = (-3u^5+u^2+u^6-2u^4+1)(1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)
d = u(u^4+3u+u^6-2u^2+1)(1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)
e = u(u^4-3u+u^6-2u^2+1)(1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)
f = (3u^5+u^2+u^6-2u^4+1)(1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)
g = (-3u^5+u^2+u^6-2u^4+1)(-1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)
h = u(u^4+3u+u^6-2u^2+1)(-1+24u^5-72u^9-192u^11+72u^7+24u^23-72u^19+72u^21-192u^17+288u^15+288u^13)

u is arbitrary.