Question 51
Define a function $f : \mathbb{N} \to \mathbb{N}$ by $f(1) = 1$, $f(n + 1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.
(This problem was suggested by Evan O'Dorney.)
(This problem was suggested by Evan O'Dorney.)