1.Introduction

We show if n is even and n > 5 then X1^5 + X2^3 +...+ Xn^5=0 has infinitely many integer solutions.
     
     
2.Theorem
     
When n is even and n > 5 then X1^5 + X2^3 +...+ Xn^5=0 has infinitely many integer solutions.
 
 
Proof.

X1^5 + X2^3 +...+ Xn^5=0...............................................(1)

First, we teat the case for n=6.

X1^5 + X2^5 + X3^5 + X4^5 + X5^5 + X6^5=0..............................(2)

Let X1=pt+a, X2=qt+b, X3=t+1, X4=-pt-b, X5=-qt-1, X6=-t-a. 

(5bq^4+5-5bp^4-5q^4+5ap^4-5a)t^4
+(10a^2p^3+10b^2q^3+10-10b^2p^3-10q^3-10a^2)t^3
+(10a^3p^2+10b^3q^2+10-10b^3p^2-10q^2-10a^3)t^2
+(5a^4p-5a^4+5b^4q+5-5b^4p-5q)t=0......................................(3)

We can obtain a parametric solution by the following method.

Next, we we teat the case for n=8.

Let X1=pt+a, X2=pt+b, X3=qt+c, X4=t+1, X5=-pt-c, X6=-pt-1, X7=-qt-a, X8=-t-b. 

Similarly, we can obtain a parametric solution by the following method.
Since parametric solution has many terms, we omit it.

Next, we we teat the case for n=10.

Let X1=pt+a, X2=pt+b, X3=qt+c, X4=qt+d, X5=t+1, X6=-pt-c, X7=-pt-d, X8=-qt-1, X9=-qt-a, X10=-t-b. 

Similarly, we can obtain a parametric solution by the following method.



Generally, equation (1) becomes to below
C4t^4 + C3t^3 + C2t^2 + C1t=0..........................................(4)
Cn: coefficient of t^n

Since equation (4) has a trivial solution {p,q}={1,1}, we can always obtain a rational solution {p,q,t} of
simultaneous equation {C2=0, C1=0, C4t^4 + C3t^3=0}.

Thus we can obtain a parametric solution of (1) if n is even and n > 5.

Consequently, equation (1) has infinitely many integer solutions if n is even and n > 5.


3.Example

We show a parametric solution for n=6.

X1=(b^6+b^5+2*b^4+b^3+2*b^2+b+1)*a^18+(b^7+4*b^6+5*b^5+5*b^4+5*b^3+5*b^2+4*b+1)*a^17
  +(2*b^8+3*b^7+11*b^6+6*b^5+9*b^4+6*b^3+11*b^2+3*b+2)*a^16+(2*b^9+4*b^8+8*b^7+14*b^6+8*b^5+8*b^4+14*b^3+8*b^2+4*b+2)*a^15
  +(b^10-3*b^9-3*b^7+5*b^6-4*b^5+15*b^4+11*b^3+12*b^2+3*b+3)*a^14+(2*b^11-7*b^10-15*b^9-17*b^8-23*b^7-16*b^6-4*b^5+7*b^4+13*b^3+3*b^2-b+2)*a^13
  +(b^12-7*b^11-22*b^10-34*b^9-36*b^8-37*b^7-20*b^6+b^5+16*b^4+10*b^3-2*b^2-b+3)*a^12+(2*b^13-7*b^12-15*b^11-32*b^10-42*b^9-35*b^8-43*b^7-29*b^6-5*b^5-6*b^4-12*b^3-9*b^2-b+2)*a^11
  +(b^14-3*b^13-10*b^12-16*b^11-24*b^10-16*b^9-21*b^8-18*b^7-27*b^6+4*b^5-24*b^4-8*b^3-2*b^2+3*b+3)*a^10+(2*b^15+b^14+9*b^13+6*b^12+14*b^11+24*b^10+14*b^9+10*b^8-22*b^7-26*b^6-28*b^5-34*b^4-10*b^3+3*b^2+7*b+2)*a^9
  +(4*b^14+11*b^13+20*b^12+37*b^11+51*b^10+46*b^9+8*b^8+4*b^7-47*b^6-27*b^5-40*b^4-5*b^3+12*b^2+6*b+2)*a^8+(b^17+b^16+8*b^15+9*b^14+29*b^13+39*b^12+61*b^11+78*b^10+38*b^9+28*b^8-12*b^7-27*b^6-25*b^5-19*b^4+17*b^3+14*b^2+7*b+1)*a^7
  +(-b^18-2*b^17-5*b^16-6*b^15+b^14+20*b^13+56*b^12+75*b^11+77*b^10+38*b^9-15*b^8-27*b^7-54*b^6-32*b^5+b^4+14*b^3+15*b^2+4*b+1)*a^6+(-3*b^18-7*b^17-18*b^16-16*b^15-12*b^14+28*b^13+57*b^12+67*b^11+60*b^10-12*b^9-27*b^8-45*b^7-36*b^6+8*b^5+20*b^4+26*b^3+11*b^2+3*b)*a^5
  +(-6*b^18-15*b^17-27*b^16-28*b^15-5*b^14+27*b^13+64*b^12+58*b^11+24*b^10-26*b^9-48*b^8-51*b^7-15*b^6+8*b^5+27*b^4+15*b^3+6*b^2)*a^4+(-7*b^18-15*b^17-26*b^16-22*b^15-13*b^14+13*b^13+26*b^12+8*b^11-4*b^10-34*b^9-33*b^8-19*b^7-2*b^6+18*b^5+15*b^4+7*b^3)*a^3
  +(-6*b^18-11*b^17-15*b^16-16*b^15-4*b^14+7*b^13+10*b^12+3*b^11-2*b^10-17*b^9-8*b^8-10*b^7+5*b^6+7*b^5+6*b^4)*a^2+(-3*b^18-4*b^17-7*b^16-8*b^15-5*b^14-5*b^13-5*b^12-5*b^11-5*b^10-5*b^9-2*b^8-b^7+2*b^6+3*b^5)*a+b^6-2*b^9-2*b^13-b^12-b^17-b^14-2*b^15-2*b^16-b^7-b^18-b^10-2*b^11
  
X2=(b^6+3*b^5+6*b^4+7*b^3+6*b^2+3*b+1)*a^18+(b^7+4*b^6+11*b^5+15*b^4+15*b^3+7*b^2+2*b-1)*a^17
  +(2*b^8+7*b^7+15*b^6+26*b^5+27*b^4+18*b^3+5*b^2-b)*a^16+(2*b^9+6*b^8+14*b^7+14*b^6+20*b^5+8*b^4-2*b^3-10*b^2-2*b-2)*a^15
  +(3*b^10+7*b^9+12*b^8+17*b^7+b^6+8*b^5-15*b^4-19*b^3-8*b^2-5*b-1)*a^14+(2*b^11+3*b^10+3*b^9-5*b^8-19*b^7-32*b^6-36*b^5-51*b^4-33*b^3-17*b^2-5*b-2)*a^13
  +(3*b^12-b^11-2*b^10-10*b^9-40*b^8-25*b^7-54*b^6-45*b^5-48*b^4-34*b^3-2*b^2-5*b-1)*a^12+(2*b^13-b^12-9*b^11-8*b^10-34*b^9-27*b^8-27*b^7-27*b^6-27*b^5-26*b^4-4*b^3+3*b^2-5*b-2)*a^11
  +(3*b^14-b^13-2*b^12-12*b^11-24*b^10-28*b^9-47*b^8-12*b^7-15*b^6-12*b^5+24*b^4+8*b^3+10*b^2-5*b-1)*a^10+(2*b^15+3*b^14+3*b^13+10*b^12-6*b^11+4*b^10-26*b^9+4*b^8+28*b^7+38*b^6+60*b^5+58*b^4+26*b^3+7*b^2-5*b-2)*a^9
  +(2*b^16+4*b^15+12*b^14+13*b^13+16*b^12-5*b^11-27*b^10-22*b^9+8*b^8+38*b^7+77*b^6+67*b^5+64*b^4+13*b^3-4*b^2-8*b-2)*a^8+(b^17+3*b^16+8*b^15+11*b^14+7*b^13+b^12-29*b^11-18*b^10+10*b^9+46*b^8+78*b^7+75*b^6+57*b^5+27*b^4-13*b^3-16*b^2-7*b-1)*a^7
  +(b^18+4*b^17+11*b^16+14*b^15+15*b^14-4*b^13-20*b^12-43*b^11-21*b^10+14*b^9+51*b^8+61*b^7+56*b^6+28*b^5-5*b^4-22*b^3-15*b^2-4*b-1)*a^6+(b^18+5*b^17+6*b^16+8*b^15-4*b^14-16*b^13-37*b^12-35*b^11-16*b^10+24*b^9+37*b^8+39*b^7+20*b^6-12*b^5-28*b^4-26*b^3-11*b^2-3*b)*a^5
  +(2*b^18+5*b^17+9*b^16+8*b^15+5*b^14-23*b^13-36*b^12-42*b^11-24*b^10+14*b^9+20*b^8+29*b^7+b^6-16*b^5-27*b^4-15*b^3-6*b^2)*a^4+(b^18+5*b^17+6*b^16+14*b^15-3*b^14-17*b^13-34*b^12-32*b^11-16*b^10+6*b^9+11*b^8+9*b^7-6*b^6-18*b^5-15*b^4-7*b^3)*a^3
  +(2*b^18+5*b^17+11*b^16+8*b^15-15*b^13-22*b^12-15*b^11-10*b^10+9*b^9+4*b^8+8*b^7-5*b^6-7*b^5-6*b^4)*a^2+(b^18+4*b^17+3*b^16+4*b^15-3*b^14-7*b^13-7*b^12-7*b^11-3*b^10+b^9+b^7-2*b^6-3*b^5)*a+2*b^9+b^14+b^10-b^6+2*b^13+b^18+2*b^15+2*b^16+b^17+b^7+2*b^11+b^12
  
X3=(-3*b-3*b^5-1-6*b^2-7*b^3-6*b^4-b^6)*a^18+(1-b^7-15*b^4-11*b^5-15*b^3-7*b^2-4*b^6-2*b)*a^17
  +(-7*b^7-18*b^3-27*b^4-15*b^6+b-26*b^5-2*b^8-5*b^2)*a^16+(-28*b^5-16*b^7-8*b^8+8*b^2+2-2*b^9-16*b^4-6*b^3-22*b^6)*a^15
  +(b-b^10+b^4-5*b^6+4*b^2-12*b^5-4*b^8-13*b^7+9*b^3+1-5*b^9)*a^14+(11*b^3+28*b^6+2-3*b+9*b^2-5*b^10-2*b^11+13*b^8+27*b^7+20*b^5+29*b^4+7*b^9)*a^13
  +(6*b^3+1-5*b^11-7*b+56*b^6+64*b^8+57*b^7-b^12-10*b^2+20*b^4+10*b^10+39*b^5+26*b^9)*a^12+(-7*b+3*b^11+58*b^9+75*b^7+14*b^4-15*b^2+8*b^10-16*b^3+61*b^6+37*b^5+2-5*b^12+67*b^8-2*b^13)*a^11
  +(-22*b^2-4*b^11-7*b-2*b^12+78*b^7-b^14-32*b^3+77*b^8-5*b^13+51*b^6+60*b^9-24*b^4+1+24*b^10+24*b^5)*a^10+(-12*b^10+38*b^8-34*b^12-26*b^11+46*b^7+2-17*b^13+14*b^6+38*b^9-3*b-42*b^4-15*b^2-5*b^14-16*b^5-2*b^15-34*b^3)*a^9
  +(2-33*b^13-48*b^12-2*b^15+8*b^8-8*b^14-36*b^4-17*b^3+10*b^7-15*b^10-21*b^6-27*b^11+28*b^9-35*b^5+4*b)*a^8+(-37*b^5-23*b^4-51*b^13+8*b^2-3*b^3+4*b^9-18*b^7-10*b^15+1+3*b-12*b^10-45*b^12-27*b^11-43*b^6-22*b^8-b^16-19*b^14-b^17)*a^7
  +(-36*b^13-15*b^14+4*b+5*b^4+1+11*b^2+14*b^3-29*b^7-16*b^5-47*b^10-2*b^15-27*b^11-26*b^9-27*b^8-20*b^6+5*b^16+b^18-54*b^12+2*b^17)*a^6+(18*b^16+3*b^18+5*b^2-4*b^5+6*b^3+b+8*b^4+8*b^14-4*b^6+7*b^17+b^7+4*b^9-27*b^11-5*b^8+8*b^15-32*b^13-28*b^10-25*b^12)*a^5
  +(15*b^6+8*b^5+5*b^3-34*b^11+15*b^17+6*b^18+b^14-24*b^10+9*b^4+27*b^16+7*b^7-6*b^9-40*b^12+2*b^2+16*b^8+20*b^15-19*b^13)*a^4+(10*b^9+14*b^15+14*b^6+17*b^14+13*b^8+5*b^4+6*b^5+15*b^17-8*b^11-10*b^12+11*b^7-5*b^13+7*b^18+b^3+26*b^16-12*b^10)*a^3
  +(15*b^16+3*b^9+14*b^15-2*b^12-9*b^11-2*b^10+6*b^18+12*b^8+11*b^6+2*b^4+12*b^14+3*b^13+8*b^7+5*b^5+11*b^17)*a^2+(-b^11+7*b^14-b^12+3*b^18-b^10+3*b^9+3*b^13+6*b^15+4*b^17+4*b^8+b^5+4*b^6+7*b^16+3*b^7)*a+b^7+3*b^14+2*b^16+b^17+b^18+2*b^15+2*b^8+2*b^9+3*b^10+b^6+2*b^11+3*b^12+2*b^13

X4=(b^6+3*b^5+6*b^4+7*b^3+6*b^2+3*b+1)*a^18+(-b^7+2*b^6+7*b^5+15*b^4+15*b^3+11*b^2+4*b+1)*a^17
  +(-b^7+5*b^6+18*b^5+27*b^4+26*b^3+15*b^2+7*b+2)*a^16+(-2*b^9-8*b^7+6*b^6+16*b^5+28*b^4+22*b^3+16*b^2+8*b+2)*a^15
  +(-b^10-b^9-4*b^8-9*b^7-b^6+12*b^5+5*b^4+13*b^3+4*b^2+5*b+1)*a^14+(-2*b^11+3*b^10-9*b^9-11*b^8-29*b^7-20*b^6-28*b^5-27*b^4-13*b^3-7*b^2+5*b+2)*a^13
  +(-b^12+7*b^11+10*b^10-6*b^9-20*b^8-39*b^7-56*b^6-57*b^5-64*b^4-26*b^3-10*b^2+5*b+1)*a^12+(-2*b^13+7*b^12+15*b^11+16*b^10-14*b^9-37*b^8-61*b^7-75*b^6-67*b^5-58*b^4-8*b^3-3*b^2+5*b+2)*a^11
  +(-b^14+7*b^13+22*b^12+32*b^11+24*b^10-24*b^9-51*b^8-78*b^7-77*b^6-60*b^5-24*b^4+4*b^3+2*b^2+5*b+1)*a^10+(-2*b^15+3*b^14+15*b^13+34*b^12+42*b^11+16*b^10-14*b^9-46*b^8-38*b^7-38*b^6+12*b^5+26*b^4+34*b^3+17*b^2+5*b+2)*a^9
  +(-2*b^16-4*b^15+17*b^13+36*b^12+35*b^11+21*b^10-10*b^9-8*b^8-28*b^7+15*b^6+27*b^5+48*b^4+33*b^3+8*b^2+2*b)*a^8+(-b^17-3*b^16-8*b^15+3*b^14+23*b^13+37*b^12+43*b^11+18*b^10+22*b^9-4*b^8+12*b^7+27*b^6+45*b^5+51*b^4+19*b^3+10*b^2+b+1)*a^7
  +(-b^18-4*b^17-11*b^16-14*b^15-5*b^14+16*b^13+20*b^12+29*b^11+27*b^10+26*b^9+47*b^8+27*b^7+54*b^6+36*b^5+15*b^4+2*b^3-5*b^2-2*b-1)*a^6+(-b^18-5*b^17-6*b^16-8*b^15+4*b^14+4*b^13-b^12+5*b^11-4*b^10+28*b^9+27*b^8+25*b^7+32*b^6-8*b^5-8*b^4-18*b^3-7*b^2-3*b)*a^5
  +(-2*b^18-5*b^17-9*b^16-8*b^15-15*b^14-7*b^13-16*b^12+6*b^11+24*b^10+34*b^9+40*b^8+19*b^7-b^6-20*b^5-27*b^4-15*b^3-6*b^2)*a^4+(-b^18-5*b^17-6*b^16-14*b^15-11*b^14-13*b^13-10*b^12+12*b^11+8*b^10+10*b^9+5*b^8-17*b^7-14*b^6-26*b^5-15*b^4-7*b^3)*a^3
  +(-2*b^18-5*b^17-11*b^16-8*b^15-12*b^14-3*b^13+2*b^12+9*b^11+2*b^10-3*b^9-12*b^8-14*b^7-15*b^6-11*b^5-6*b^4)*a^2+(-b^18-4*b^17-3*b^16-4*b^15-3*b^14+b^13+b^12+b^11-3*b^10-7*b^9-6*b^8-7*b^7-4*b^6-3*b^5)*a-3*b^14-2*b^8-2*b^9-3*b^10-b^18-2*b^15-2*b^16-b^17-b^7-b^6-2*b^11-3*b^12-2*b^13

X5=(-3*b-3*b^5-1-6*b^2-7*b^3-6*b^4-b^6)*a^18+(-1+b^7-15*b^4-2*b^6-7*b^5-11*b^2-4*b-15*b^3)*a^17
  +(-26*b^3-2-15*b^2+b^7-7*b-5*b^6-18*b^5-27*b^4)*a^16+(-6*b-8*b^5+10*b^7-14*b^2-2+2*b^9-20*b^4+2*b^8+2*b^6-14*b^3)*a^15
  +(5*b^9-12*b^2+b^10+15*b^6-17*b^3+8*b^8-8*b^5-3+19*b^7-b^4-7*b)*a^14+(2*b^11-3*b^2+19*b^4+33*b^8-3*b+36*b^6+32*b^5+17*b^9+5*b^3+5*b^10-2+51*b^7)*a^13
  +(b+5*b^11+10*b^3+25*b^5+2*b^2+40*b^4+34*b^9+2*b^10-3+b^12+48*b^8+54*b^6+45*b^7)*a^12+(8*b^3+9*b^2+b+27*b^5+27*b^8+34*b^4+4*b^10+2*b^13+26*b^9+27*b^7-3*b^11+5*b^12+27*b^6-2)*a^11
  +(12*b^7+15*b^8+12*b^9+2*b^2-8*b^11-10*b^12+b+24*b^4+47*b^6-3+28*b^5+5*b^13+12*b^3+b^14-24*b^10)*a^10+(26*b^6-4*b^7-7*b^13+6*b^4-38*b^9+5*b^14-60*b^10-58*b^11-26*b^12-10*b^3-2-28*b^8-3*b^2+2*b^15-4*b^5-3*b)*a^9
  +(-8*b^8+22*b^7+2*b^16-2-64*b^12+8*b^15-67*b^11-77*b^10-16*b^4-38*b^9-13*b^13+5*b^5+27*b^6-4*b-13*b^3-12*b^2+4*b^14)*a^8+(-57*b^12-7*b^4-27*b^13+18*b^7+16*b^15-78*b^10-b^5-1-11*b^3-75*b^11-8*b^2+29*b^6-3*b+13*b^14-46*b^9-10*b^8+7*b^16+b^17)*a^7
  +(-11*b^2-4*b-1+43*b^7+5*b^14+21*b^8+22*b^15+15*b^16-51*b^10+4*b^5-61*b^11-28*b^13+b^18+20*b^6+4*b^17-14*b^9-14*b^3-15*b^4-56*b^12)*a^6+(-24*b^10-5*b^2+26*b^16-8*b^4+11*b^17+3*b^18+12*b^14-37*b^11+4*b^5+35*b^8-20*b^13-6*b^3+16*b^6-b+37*b^7+28*b^15+16*b^9-39*b^12)*a^5
  +(36*b^8-8*b^5-5*b^6+27*b^16+6*b^18+23*b^7+15*b^17-20*b^12-2*b^2-5*b^3+16*b^15-29*b^13-b^14+42*b^9-14*b^11-9*b^4+24*b^10)*a^4+(17*b^8+15*b^17-14*b^6-5*b^4+3*b^7+34*b^9-11*b^13-6*b^12+6*b^15-6*b^5+32*b^10+7*b^18+18*b^16-9*b^14-b^3+16*b^11)*a^3
  +(5*b^16-8*b^15+22*b^10+15*b^11+6*b^18+15*b^9-8*b^7-11*b^6-5*b^5-2*b^4+7*b^17+10*b^12-4*b^14-9*b^13)*a^2+(7*b^11+2*b^17+7*b^10+7*b^12-b^14+3*b^18+3*b^9-4*b^8+3*b^13-b^5-4*b^6-3*b^7-b^16)*a-2*b^9-2*b^8-b^12-b^6-b^14-2*b^11-b^17-2*b^13-b^10-2*b^15-b^7+b^18

X6=(-b^5-2*b^2-b-1-b^3-2*b^4-b^6)*a^18+(-5*b^3-1-5*b^5-5*b^4-4*b-5*b^2-4*b^6-b^7)*a^17
  +(-2*b^8-3*b-11*b^2-2-6*b^3-11*b^6-6*b^5-9*b^4-3*b^7)*a^16+(-14*b^6-4*b-8*b^5-2*b^9-8*b^4-2-4*b^8-8*b^7-14*b^3-8*b^2)*a^15
  +(3*b^3-15*b^6-1-11*b^7-3*b^9-5*b^4-12*b^8-3*b^10+4*b^5+3*b)*a^14+(16*b^5+17*b^3-13*b^8-7*b^7+b^10+15*b^2-2*b^11-3*b^9+4*b^6+23*b^4+7*b-2)*a^13
  +(34*b^3+37*b^5+36*b^4+22*b^2+20*b^6-1+7*b-b^7-16*b^8+2*b^10+b^11-10*b^9-3*b^12)*a^12+(15*b^2+7*b-2+29*b^7+12*b^10-2*b^13+35*b^5+43*b^6+6*b^9+32*b^3+5*b^8+9*b^11+b^12+42*b^4)*a^11
  +(21*b^6-3*b^14+2*b^12+24*b^4+16*b^3+18*b^7+8*b^11+24*b^10+16*b^5+3*b+10*b^2-1-3*b^13-4*b^9+27*b^8)*a^10+(10*b^12-14*b^4-10*b^7-9*b^2-2-2*b^15+28*b^10+22*b^8+34*b^11-6*b^3-7*b^14-24*b^5-3*b^13-14*b^6+26*b^9-b)*a^9
  +(5*b^13+27*b^11-6*b^15+47*b^10-37*b^5-4*b^2-20*b^4-4*b^9-51*b^6-11*b^3-8*b^8+40*b^12-46*b^7-2*b^16-12*b^14)*a^8+(-9*b^3-39*b^5-7*b^16-38*b^8-29*b^4-8*b^2+12*b^10+19*b^13-b-78*b^7-17*b^14+27*b^11-1-b^17+25*b^12-61*b^6-14*b^15-28*b^9)*a^7
  +(-b^18-4*b^17+6*b^3+2*b-b^14-38*b^9+27*b^11+5*b^2-15*b^16-77*b^8-14*b^15-56*b^6-75*b^7-b^4-20*b^5+1+54*b^12+32*b^13+15*b^10)*a^6+(7*b^2-67*b^8+12*b^5-26*b^16+12*b^10+45*b^12-8*b^14+16*b^4+3*b+36*b^13-3*b^18-57*b^7+18*b^3+27*b^11-28*b^6-11*b^17-20*b^15-60*b^9)*a^5
  +(-64*b^8-15*b^17-58*b^9+5*b^6+48*b^12+28*b^5+15*b^14+26*b^11-24*b^10-27*b^16+51*b^13+27*b^4-27*b^7-6*b^18+6*b^2+15*b^3-8*b^15)*a^4+(2*b^15-13*b^8-15*b^17+13*b^7+7*b^3+22*b^6+34*b^12+4*b^11+15*b^4-8*b^10+26*b^5+19*b^14-26*b^9+33*b^13-18*b^16-7*b^18)*a^3
  +(16*b^7-3*b^11-7*b^9+8*b^14+11*b^5-6*b^18+10*b^15+2*b^12-7*b^17-5*b^16+15*b^6+17*b^13+4*b^8+6*b^4-10*b^10)*a^2+(5*b^14+7*b^7+b^16-2*b^17+5*b^13+5*b^10+2*b^15+8*b^8-3*b^18+3*b^5+4*b^6+5*b^9+5*b^11+5*b^12)*a+b^7+2*b^9+b^10+2*b^11+b^12+2*b^13+b^14+2*b^15+b^17-b^18+b^6+2*b^8

Numeric examples:
(a,b)=(3,2)
[X1,X2,X3,X4,X5,X6]=[13636793, 36873121, -36231625, 16280575, -6955753, -23603111]
(a,b)=(3,-2)
[X1,X2,X3,X4,X5,X6]=[43372767, -4183929, 11359729, 65315033, -61028751, -54834849]