1.Introduction

We show that simultaneous equation {a^3+b^3+c^3 = d^3+e^3+f^3 , abc=def} has infinitely many parametric solutions.
                 
                 
2.Theorem
     
Simultaneous diophantine equation {a^3+b^3+c^3 = d^3+e^3+f^3 , abc=def} has infinitely many parametric solutions.

condition: 
p,q are arbitrary.

 
Proof.

a^3+b^3+c^3 = d^3+e^3+f^3...............................................................................................(1)

Let a = p(x+s), b = q(x+t), c = x, d = px, e = q(x+s), f = x+t the equation (1) becomes to below equation.

(-3t-3q^3s+3p^3s+3q^3t)x^2+(-3q^3s^2+3q^3t^2-3t^2+3p^3s^2)x+p^3s^3-t^3-q^3s^3+q^3t^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 = (6p^3q^3-3q^6-3p^6)U^4+(12q^6+12p^3-12q^3-12p^3q^3)U^3+(18p^3q^3-18q^6-18p^3+18q^3)U^2
    +(12q^6+12p^3-12q^3-12p^3q^3)U-3+6q^3-3q^6..........................................................................(3)
    
Since quartic equation (3) has a rational point Q(U,V)=(-(-1+q^3)/(-q^3+p^3),3(-1+q)(q^2+q+1)(p-1)(p^2+p+1)/((-q+p)(p^2+pq+q^2))), then this quartic equation is birationally equivalent to an elliptic curve below.

Y^2+(-4q^3+8-4p^3)YX = X^3+(-4q^6+10p^3q^3-2q^3-4p^6+2-2p^3)X^2
                         +(108p^6-216p^3-216p^6q^3+432p^3q^3+108-216q^3+108p^6q^6-216p^3q^6+108q^6)X
                         -1512p^6q^9+216-432p^6q^12+864p^12q^3-216p^9q^3+1080p^9q^9-432p^12q^6-432q^12
                         -648p^3+2808p^3q^3+864p^3q^12-1512p^9q^6+648p^9+216q^6+216p^6-648q^3+648q^9-432p^12-2808p^3q^6-216p^3q^9+4536p^6q^6-2808p^6q^3
Transformation is given, 

U = (24p^3q^9-24q^9-60p^6q^6+48p^3q^6+12q^6+24p^9q^3+48p^6q^3-96p^3q^3-6p^3q^3X+24q^3+q^3Y+6q^3X-24p^9+12p^6+6p^3X+24p^3-Y-6X-12)/(q^3Y-Yp^3)
V = (648-1296p^9q^15-120Yp^3-3888p^3q^15-25920p^6q^9+6480p^9q^3+25920p^9q^9+48p^3q^12Y-3888p^15q^3-6480p^9q^12+996Xp^6+6480p^12q^3+3888p^15q^6
    -6480p^12q^9+3888p^6q^15-1296p^15q^9+3240p^12q^12+1548q^6Xp^6+996q^6X-3q^3X^3p^3+36q^3X^2-120q^3Y-2508q^3Xp^6-2508q^6p^3X-360q^6Yp^3+480q^3Yp^3
    -2592p^3-96Xp^3q^15+480Xp^6q^12-468Xp^9q^9-96Xp^15q^3+480Xp^12q^6-480p^12q^3X-492p^9X-3240q^12-19440p^6q^3-36p^9X^2-516p^6q^9X+12960p^3q^3+1296q^15
    +6480p^3q^12+96p^15X-25920p^9q^6+36p^9q^3X^2+1296p^9-492q^9X+96q^15X+1476q^9Xp^3+18q^6X^2+1476q^3Xp^9-516p^9Xq^6-480q^12Xp^3+2592q^6+2592p^6-2592q^3
    +1296q^9-3240p^12+1296p^15-90p^6q^6X^2+18p^6X^2-19440p^3q^6+6480p^3q^9+38880p^6q^6+300X+48Y-3X^3-18X^2+72q^6X^2p^3-36q^9X^2+72p^6q^3X^2+36p^3q^9X^2+3X^3p^3
    -144X^2p^3q^3+3X^3q^3+48p^12q^3Y-48p^12Y+2508p^3q^3X-900p^3X-900q^3X-72p^6q^9Y+120q^9Y-48q^12Y+120p^9Y-72p^9q^6Y-48p^3q^9Y+432p^6q^6Y-360p^6q^3Y-48p^9q^3Y+36X^2p^3)/(q^3Y^2-p^3Y^2).

The point corresponding to point Q is P(X,Y)=(4q^6-10p^3q^3+2q^3+4p^6-2+2p^3, 16q^9-24p^3q^6-24q^6+96p^3q^3-24q^3-24p^6q^3-24p^6+16+16p^9-24p^3).

Hence we get 2P(X,Y)=(2(-10+80q^18-393p^12q^6-583p^12q^3+40p^3+4p^21+507p^15q^3+507p^3q^15-583p^3q^12-393p^6q^12+1192p^6q^9-13p^9q^3+2p^24+2q^24
                     +4q^21-13p^3q^9-347p^15q^9+167q^12p^9+57p^15q^6+392p^12q^12+167p^12q^9-347p^9q^15-188p^18q^3-20q^21p^3+80p^18+723p^6q^3-188q^18p^3
                     +57q^15p^6+723p^3q^6-1788p^6q^6+164p^18q^6+164q^18p^6-20p^21q^3+1192p^9q^6-329q^15-668p^9q^9-443p^9-482p^3q^3+557p^12+101p^6+40q^3
                     +101q^6-443q^9+557q^12-329p^15)/((q^3+1-2p^3)^2(2q^3-1-p^3)^2(q^3-2+p^3)^2),
                      24(q^6-p^3q^3-q^3-p^3+p^6+1)(3p^3+3p^6q^3-6p^3q^6+6p^3q^3-6p^6-6q^3+3q^6+q^9+p^9+1)(-6p^3-6p^6q^3+3p^3q^6+6p^3q^3+3p^6+3q^3-6q^6+q^9+p^9+1)
                      (q^12+q^9-5p^3q^9-15p^3q^6+6q^6+15p^6q^6-14q^3-15p^6q^3-5p^9q^3+30p^3q^3+7+p^12-14p^3+p^9+6p^6)/((q^3+1-2p^3)^3(2q^3-1-p^3)^3(q^3-2+p^3)^3)).

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.Example

m=2:

a = p(p^3-2+q^3)(p^12+10p^9-14p^9q^3-6p^6q^3+24p^6q^6-12p^6+4p^3+12p^3q^3-6p^3q^6-14p^3q^9+10q^9+4q^3-12q^6-2+q^12)
    
b = (2p^3-1-q^3)(2p^12-4p^9-4p^9q^3-12p^6q^3+12p^6q^6+12p^6-10p^3+6p^3q^3+6p^3q^6-10p^3q^9+14q^9+14q^3-24q^6-1-q^12)
    
c = (p^3+1-2q^3)(p^12-14p^9+10p^9q^3+24p^6-6p^6q^3-12p^6q^6-14p^3+12p^3q^6-6p^3q^3+4p^3q^9+4q^9+10q^3-12q^6+1-2q^12)
 
d = p(p^3+1-2q^3)(p^12-14p^9+10p^9q^3+24p^6-6p^6q^3-12p^6q^6-14p^3+12p^3q^6-6p^3q^3+4p^3q^9+4q^9+10q^3-12q^6+1-2q^12)
    
e = (p^3-2+q^3)(p^12+10p^9-14p^9q^3-6p^6q^3+24p^6q^6-12p^6+4p^3+12p^3q^3-6p^3q^6-14p^3q^9+10q^9+4q^3-12q^6-2+q^12)
    
f = (2p^3-1-q^3)(2p^12-4p^9-4p^9q^3-12p^6q^3+12p^6q^6+12p^6-10p^3+6p^3q^3+6p^3q^6-10p^3q^9+14q^9+14q^3-24q^6-1-q^12)