6 papers
On Zeckendorf-Niven numbers and arithmetic progressions
Kelly Lao, Steven J. Miller, Nicholas Rosa +4
A positive integer is Zeckendorf-Niven (respectively, Lucas-Niven) if it is divisible by the number of summands in its Zeckendorf decomposition (respectively, Lucas decomposition).…
Integers Having in Both Zeckendorf And Chung-Graham Decompositions
Lucas Bustos, Hung Viet Chu, Minchae Kim +3
Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positiv…
On a Roll Again: Analysis of a Dice Removal Game
Francesco Camellini, Wissam Ghantous, Andrea M. Lanocita +4
Suppose we have dice, each with faces (assume ). On the first turn, roll all of them, and remove from play those that rolled an . Roll all of the remaining dice…
Problems Regarding a Pair of Diophantine Equations
Hung Viet Chu, Steven J. Miller, Garrett Tresch
For two relatively prime positive integers , it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1…
General Recurrence Multidimensional Zeckendorf Representations
Jiarui Cheng, Steven J. Miller, Sebastian Rodriguez-Labastida +3
We present a multidimensional generalization of Zeckendorf's Theorem (any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers) to a large family of…
Linear Recurrences from Counting Schreier-Type Multisets
Hung Viet Chu, Yubo Geng, Julian King +3
A nonempty set is Schreier if . Bird observed that counting Schreier sets in a certain way produces the Fibonacci sequence. Since then, various connections betwe…