1.Introduction

About X^n + Y^n + Z^n = Square number,n=1,2,3.
We show this equation has infinitely many integer solutions.

2.Theorem
     
A simultaneous equation {x + y + z = u^2, x^2 + y^2 + z^2 = v^2, x^3 + y^3 + z^3 = w^2} has infinitely many integer solutions.
 
Proof.

x + y + z = u^2....................................................................(1)
x^2 + y^2 + z^2 = v^2..............................................................(2)
x^3 + y^3 + z^3 = w^2..............................................................(3)

First, substitute z=u^2-x-y to equation (3) then (3) becomes to (4).

(3u^2-3y)x^2+(-3y^2+6u^2y-3u^4)x+3u^2y^2-3u^4y+u^6 = w^2...........................(4)

From equation (4), we obtain a parametric solution for (x,y,z,w) as below.

x = (3u^2a-ap^2+3u^4+2u^3p)/(3u^2+3a-p^2)
y = -a
z = -(-3u^2a+u^2p^2-3a^2+2u^3p)/(3u^2+3a-p^2)
w = (3u^5+3u^3a+u^3p^2-3pa^2+3pu^4)/(3u^2+3a-p^2).

Then equation (2) becomes to equation (5).

v^2 = (27u^4a^2+18u^6a+2a^2p^4+12u^7p+8u^6p^2+36a^3u^2-6a^3p^2
    + u^4p^4+4u^5p^3-18u^2a^2p^2-12ap^2u^4-4ap^3u^3-12a^2u^3p
    +9u^8+18a^4)/((3u^2+3a-p^2)^2).................................................(5)


Set U = p, then V = (3u^2+3a-p^2)v.
V^2 = (2a^2+u^4)U^4+(-4u^3a+4u^5)U^3+(-6a^3+8u^6-12u^4a-18u^2a^2)U^2
    + (12u^7-12a^2u^3)U+27u^4a^2+18u^6a+18a^4+36a^3u^2+9u^8........................(6)
    
Set (a,u)=(2,1), this quartic equation V^2=9p^4-4p^3-136p^2-36p+729 is birationally equivalent to an elliptic curve below.

Y^2 = X^3-X^2-32265X+ 2233737......................................................(7)

By using Cremona's mwrank(SAGE), we obtain the rank=1 and generator is P(117 , 216) on (7).

Hence, elliptic curve (7) has infinitely many rational solutions.

Transformation is given,
U = (162X-14814)/(3Y+2X+414)
V = (-96696Y-66663X^2+7884837X-368635617+243X^3)/((3Y+2X+414)^2)
X = (54V+1458-36U-45U^2)/(U^2)
Y = (2916V+78732-2916U-7344U^2-36VU-108U^3)/(U^3).

For example, the case of P(117 , 216) on (7), corresponding point is (U,V)=(115/36, 827/144) on (7).

Then we obtain p=115/36 and v = 7443/1561.

Accordingly, integer solution is obtained such that (x,y,z)=(10155866, -4873442, -2845703).

Small solutions are shown below.
      [x , y , z]
 P    [10155866, -4873442, -2845703]
2P    [5675744542575759273577, -3131648224907118878882, -978272205215080955254]
3P    [94826306439776303571736616557558934627836562, -181346451870573104294497320322003639773806402, 177193371366083352870009363925446525032873041]

Q.E.D.