Question 45
Determine all functions $f: \mathbb{N}^* \to \mathbb{N}$ that satisfy the following properties:
a) the expression $n^3 - n f(n) - 1$ is a cube of a natural number for all $n \in \mathbb{N}^*$;
b) the function $g: \mathbb{N}^* \to \mathbb{N}$ defined by $g(n) = 4n - f(n)$ is increasing.
a) the expression $n^3 - n f(n) - 1$ is a cube of a natural number for all $n \in \mathbb{N}^*$;
b) the function $g: \mathbb{N}^* \to \mathbb{N}$ defined by $g(n) = 4n - f(n)$ is increasing.