Question 23
(YUG 2) Prove that for every natural number $k$ ($k\ge2$) there exists an irrational number $r$ such that for every natural number $m$,
$$[r^m]\equiv-1\pmod k.$$
Remark. An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.
$$[r^m]\equiv-1\pmod k.$$
Remark. An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.