Special M

 Importance: Medium ✭✭
 Author(s): Kimberling, Clark
 Subject: Number Theory
 Keywords:
 Recomm. for undergrads: yes
 Prize: \$50 (Paid by Clark Kimberling)
 Posted by: vprusso on: June 12th, 2012

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 ?)

Bibliography

http://faculty.evansville.edu/ck6/integer/unsolved.html

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