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arXiv · 2512.03468

Cyclotomic Congruences and Lucas Sequences

Tyler Ross·Zhongyan Shen·Tianxin Cai
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Abstract

In this paper, we extend the $p$-adic valuations originally obtained by Carmichael for the sequences obtained by applying M\"obius inversion to Lucas sequences to $p$-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.

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Tyler Ross, Zhongyan Shen, Tianxin Cai. 2025-12-03. Cyclotomic Congruences and Lucas Sequences. https://arxiv.org/abs/2512.03468

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