1.Introduction

We show that simultaneous equation {a^4+b^4+c^4 = d^4+e^4+f^4 ,abc=def} has infinitely many parametric solutions.
We use Stephane Vandemergel's result below.
{a^4+b^4+c^4 = d^4+e^4+f^4 ,abc=def} has a follwing parametric solution.
{a,b,c}={rp, wp, q^2},{d,e,f}={rq, wq, p^2} where p^4+q^4=r^4+w^4.(Guy's book[1])

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 ,abc=def} has infinitely many parametric solutions.

 
Proof.

According to Stephane Vandemergel's, simultaneous diophantine equation {a^4+b^4+c^4 = d^4+e^4+f^4 ,abc=def}  has a follwing parametric solution.

{a,b,c}={rp, wp, q^2},{d,e,f}={rq, wq, p^2}.
Condition: p^4 + q^ 4 = r^4 + w^4.

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 ,abc=def} has infinitely many parametric solutions.


Q.E.D.


4.Examples

a = (u^6-3u^5-2u^4+u^2+1)(u^6+u^4-2u^2-3u+1)u
b = (u^6+u^4-2u^2+3u+1)(u^6+u^4-2u^2-3u+1)u^2
c = (u^6+3u^5-2u^4+u^2+1)^2
d = (u^6-3u^5-2u^4+u^2+1)(u^6+3u^5-2u^4+u^2+1)
e = (u^6+u^4-2u^2+3u+1)(u^6+3u^5-2u^4+u^2+1)u
f = (u^6+u^4-2u^2-3u+1)^2u^2


a = (u^24+3u^23+10u^22+9u^21+4u^20-9u^19-97u^18+57u^17+136u^16-207u^15-128u^14+333u^13+179u^12-213u^11-134u^10-36u^9+43u^8+63u^7-25u^6+3u^5-2u^4+7u^2+1)
    (u^24+7u^22-2u^20+3u^19-25u^18+63u^17+43u^16-36u^15-134u^14-213u^13+179u^12+333u^11-128u^10-207u^9+136u^8+57u^7-97u^6-9u^5+4u^4+9u^3+10u^2+3u+1)u
b = (u^24+7u^22-2u^20-3u^19-25u^18-63u^17+43u^16+36u^15-134u^14+213u^13+179u^12-333u^11-128u^10+207u^9+136u^8-57u^7-97u^6+9u^5+4u^4-9u^3+10u^2-3u+1)
    (u^24+7u^22-2u^20+3u^19-25u^18+63u^17+43u^16-36u^15-134u^14-213u^13+179u^12+333u^11-128u^10-207u^9+136u^8+57u^7-97u^6-9u^5+4u^4+9u^3+10u^2+3u+1)u^2
c = (u^24-3u^23+10u^22-9u^21+4u^20+9u^19-97u^18-57u^17+136u^16+207u^15-128u^14-333u^13+179u^12+213u^11-134u^10+36u^9+43u^8-63u^7-25u^6-3u^5-2u^4+7u^2+1)^2
d = (u^24+3u^23+10u^22+9u^21+4u^20-9u^19-97u^18+57u^17+136u^16-207u^15-128u^14+333u^13+179u^12-213u^11-134u^10-36u^9+43u^8+63u^7-25u^6+3u^5-2u^4+7u^2+1)
    (u^24-3u^23+10u^22-9u^21+4u^20+9u^19-97u^18-57u^17+136u^16+207u^15-128u^14-333u^13+179u^12+213u^11-134u^10+36u^9+43u^8-63u^7-25u^6-3u^5-2u^4+7u^2+1)
e = (u^24+7u^22-2u^20-3u^19-25u^18-63u^17+43u^16+36u^15-134u^14+213u^13+179u^12-333u^11-128u^10+207u^9+136u^8-57u^7-97u^6+9u^5+4u^4-9u^3+10u^2-3u+1)
    (u^24-3u^23+10u^22-9u^21+4u^20+9u^19-97u^18-57u^17+136u^16+207u^15-128u^14-333u^13+179u^12+213u^11-134u^10+36u^9+43u^8-63u^7-25u^6-3u^5-2u^4+7u^2+1)u
f = (u^24+7u^22-2u^20+3u^19-25u^18+63u^17+43u^16-36u^15-134u^14-213u^13+179u^12+333u^11-128u^10-207u^9+136u^8+57u^7-97u^6-9u^5+4u^4+9u^3+10u^2+3u+1)^2u^2

u is arbitrary.


4.Reference

[1] Richard K. Guy: Unsolved Problems in Number Theory