Is it the only solution?

Number Theory Level 3

It is easy to find a solution for the equation below, where x , y , x, y, and z z are three non-zero integers. x 2 + y 2 + z 2 = x 3 + y 3 + z 3 x^2+y^2+z^2=x^3+y^3+z^3 This solution is x = y = z = 1 x=y=z=1 . Are there other solutions?

Yes, exactly 5 No Yes, finite many, but less than 5 Yes, infinite many Yes, finite many, but more than 5

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Áron Bán-Szabó
Jul 31, 2017

Let z = − x z=-x . Then

2 x 2 + y 2 = y 3 2 x 2 = ( y − 1 ) y 2 \begin{aligned} 2x^2+y^2 & = y^3 \\ 2x^2 & = (y-1)y^2 \end{aligned}

Let y = 2 k 2 + 1 y=2k^2+1 ( k k is an integer). Then x = k ( 2 k 2 + 1 ) x=k(2k^2+1) and the solutions are: x = k ( 2 k 2 + 1 ) y = 2 k 2 + 1 z = − k ( 2 k 2 + 1 ) \begin{aligned} x & =k(2k^2+1) \\ y & = 2k^2+1 \\ z & = -k(2k^2+1) \end{aligned}

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