Question 29
For positive integers $n$ and $k$, let $f_k(n)$ denote the remainder when $n$ is divided by $2^k - 1$. A positive integer $n$ is grouse if
$$ f_{1}(n)\leqslant f_{2}(n)\leqslant f_{3}(n)\leqslant f_{4}(n)\leqslant\cdots. $$
(a) Prove that there are infinitely many grouse numbers.
(b) Prove that there exists a positive integer M such that there are at most $ M/2025^{2025} $ grouse numbers less than M.
$$ f_{1}(n)\leqslant f_{2}(n)\leqslant f_{3}(n)\leqslant f_{4}(n)\leqslant\cdots. $$
(a) Prove that there are infinitely many grouse numbers.
(b) Prove that there exists a positive integer M such that there are at most $ M/2025^{2025} $ grouse numbers less than M.