On three conjectures of Kimberling concerning the array ⌊ kφn⌋
Abstract
Let φ be the golden ratio and let Rn={⌊ kφn⌋ : k≥ 1} be the n-th row of the array T(n,k)=⌊ kφn⌋ (OEIS A128440). In 2022 Kimberling conjectured that the rows R2n−1 and R2n are disjoint, and that after the two rows are merged and each entry is replaced by its rank, they become the lower and upper Wythoff sequences. He also conjectured (OEIS A358359) that if a(N) is the number of rows containing N, then every positive integer occurs infinitely often among the values of a. We show that the first two conjectures follow quickly from the Skolem-Bang theorem, which also yields the exact rule for when two rows are disjoint: Ri∩ Rj=∅ (i<j) if and only if j−i is odd and divides i. We then prove the third conjecture. The main tools are an explicit determination of the rows containing an odd-indexed Lucas number, which extends a result of Noppakaew, Kanwarunyu and Wanitchatchawan, and a "Lucas shift" lemma: if N+1 is not of the form L2e with e≥1, then adding a sufficiently large even-indexed Lucas number to N does not change the set of rows containing it. We also show that each value of a is taken on a set of positive natural density, and we report computations up to 108 suggesting that the least N lying in exactly v≥ 2 rows is the Lucas number L4v−5.
Related
- A Counterexample to Three Conjectures of Luo
- Solutions to Three Conjectures and an Open Problem on Binary BCH Codes
- Three Conjectures on Binary Channels for the Doubly Symmetric Binary Source
- On two problems concerning the geodesic transversal number of graphs
- Some New Sharp Boundedness Result Concerning Linear functionals on Banach spaces
BibTeX
@misc{ashburn2026three,
title = {{On three conjectures of Kimberling concerning the array $\lfloor kφ^n\rfloor$}},
author = {Alex Ashburn},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05776},
eprint = {2610.05776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05776},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On three conjectures of Kimberling concerning the array ⌊ kφ^n⌋
AU - Alex Ashburn
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05776
UR - https://arxiv.org/abs/2610.05776
AB - Let φ be the golden ratio and let R_n={⌊ kφ^n⌋ : k≥ 1} be the n-th row of the array T(n,k)=⌊ kφ^n⌋ (OEIS A128440). In 2022 Kimberling conjectured that the rows R_2n−1 and R_2n are disjoint, and that after the two rows are merged and each entry is replaced by its rank, they become the lower and upper Wythoff sequences. He also conjectured (OEIS A358359) that if a(N) is the number of rows containing N, then every positive integer occurs infinitely often among the values of a. We show that the first two conjectures follow quickly from the Skolem-Bang theorem, which also yields the exact rule for when two rows are disjoint: R_i∩ R_j=∅ (i<j) if and only if j−i is odd and divides i. We then prove the third conjecture. The main tools are an explicit determination of the rows containing an odd-indexed Lucas number, which extends a result of Noppakaew, Kanwarunyu and Wanitchatchawan, and a "Lucas shift" lemma: if N+1 is not of the form L_2e with e≥1, then adding a sufficiently large even-indexed Lucas number to N does not change the set of rows containing it. We also show that each value of a is taken on a set of positive natural density, and we report computations up to 10^8 suggesting that the least N lying in exactly v≥ 2 rows is the Lucas number L_4v−5.
ER -