Solutions of $x^3+y^3+z^3=42$

Question

I was surfing through youtube and found this amazing question on diophantine equation. If $(x,y,z) \in \mathbb{Z}$ and $$x^3+y^3+z^3=42$$ find the values of $(x,y,z)$.

I tried this equation to do by myself. All in vain. Then I decided to use softwares. All in vain. One software declared that there are no integer solutions. Even wolframalpha was not able to found the solutions.

Is there really any solution$?$ If yes how do we find it$?$ Any help is greatly appreciated.

Answer 1

$$(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 = 42$$ This was found by Andrew Booker and Andrew Sutherland using a computer search.