Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

Question

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

I am not able to find an idea on how to proceed with the above questions. I have found only the obvious solution $(1,1,1)$.

Could you please provide some hints and ideas on how to proceed with the above question? Also, can we find the solutions?

Thanks.

Answer 1

Let $z=-x$. Then $$2x^2+y^2=y^3$$ $$2x^2=(y-1)y^2$$ If $\frac{y-1}2=k^2 - $ perfect square, then $$y=2k^2+1, x=k(2k^2+1)$$ Answer: $$x=k(2k^2+1), y=2k^2+1, z=-k(2k^2+1)$$

Second method:

Let $y=1+a, z=1-a$. Then $$x^3+2(1+3a^2)=2(1+a^2)+x^2$$ $$x^2-x^3=4a^2.$$ Let $1-x=4p^2$, then $$x^2(1-x)=(4p^2-1)^24p^2=(2a)^2.$$ Let $a=p(4p^2-1)$. Then $$(x,y,z)=(1-4p^2, 1+p(4p^2-1), 1-p(4p^2-1))$$

Answer 2

Let $y=1+a, z=1-a$. Then $$x^3+2(1+3a^2)=2(1+a^2)+x^2$$ $$x^2-x^3=4a^2.$$ Let $1-x=4p^2$, then $$x^2(1-x)=(4p^2-1)^24p^2=(2a)^2.$$ Let $a=p(4p^2-1)$. Then $$(x,y,z)=(1-4p^2, 1+p(4p^2-1), 1-p(4p^2-1))$$