Cyclotomic Congruences and Lucas Sequences
Number Theory
2026-01-09 v2
Abstract
In this paper, we extend the -adic valuations originally obtained by Carmichael for the sequences obtained by applying M\"obius inversion to Lucas sequences to -adic congruences, from which we immediately derive corresponding congruences for Lucas sequences. As a corollary, we also establish some constraints on the entry point behavior of primes in Lucas sequences, on the basis of which we conjecture the presence of a strong Chebyshev-like bias in real regular Lucas sequences.
Cite
@article{arxiv.2512.03468,
title = {Cyclotomic Congruences and Lucas Sequences},
author = {Tyler Ross and Zhongyan Shen and Tianxin Cai},
journal= {arXiv preprint arXiv:2512.03468},
year = {2026}
}
Comments
21 pages, 1 table, 1 figure; replacement corrects a small but confusing typographical error in the definition of M^V