Does there exist infinitely many integer solutions for the above equation? That is, can 25 be written as the sum of 4 cubes in infinitely many ways?
This is a problem from STEMS 2023 organized by CMI(Chennai Mathematical Institute). It states "Show that 25 can be written as a sum of 4 cubes in infinitely many ways". The question doesn't state if it refers to Integral solutions.However, I've run a simple Python code and it returns around 30 solutions for the first 100 positive integers(not counting permutations of the same representation). Hence I ask, are there be infinitely many integral solutions?
Notice that $$ 25=27-2=3^3-2, \tag{1} $$ and $-2$ can be written as a sum of three cubes in infinitely many ways$^{(\dagger)}$, as $$ -2 = (-1-6c^3)^3+(-1+6c^3)^3+(6c^2)^3\quad(c\in\mathbb{Z}). \tag{2} $$ It follows from $(1)$ and $(2)$ that $25$ can be written as a sum of four cubes in infinitely many ways.
$^{(\dagger)}$ https://en.wikipedia.org/wiki/Sums_of_three_cubes.