How to prove the following statement:
"If an integer can be expressed as a sum of three non-zero cubes in a way, then it can be expressed as a sum of three non-zero cubes in infinitely many ways."?
I am not asking you to show me the proof, but hopefully one can give me a hint where I can start.
Your help would be really appreciated. THANKS!
This a part of an open problem, Sums of three cubes.
For example, parametric solutions are known for $1$ and $2$, but not for $3$.
For $3$, two representations were known for a long time,
$$ 3=1^3+1^3+1^3=4^3+4^3+(-5)^3 $$
And in $2019$, a third representation was found
$$ 3=569936821221962380720^{3}+(-569936821113563493509)^{3}+(-472715493453327032)^{3} $$
But it is not known if there are more or not.