Question 92
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square, and let $a = (1 + \sqrt{k})/2$. Show that
$$
\{\lfloor a^2 n \rfloor - \lfloor a \lfloor an \rfloor \rfloor : n = 1, 2, 3, \dots\} = \{1, \dots, \lfloor a \rfloor\}.
$$
$$
\{\lfloor a^2 n \rfloor - \lfloor a \lfloor an \rfloor \rfloor : n = 1, 2, 3, \dots\} = \{1, \dots, \lfloor a \rfloor\}.
$$