Let denote the golden ratio, and let denote the floor function. For fixed , let , let , and let . We can expect to have about the same growth rate as .
Conjecture Prove or disprove that for every fixed , as ranges through all the positive integers, there is a number such that takes each of the values infinitely many times, and . (Can you formulate as a function of ? Generalize for other numbers ?)
* indicates original appearance(s) of problem.