Cyclotomic Congruences and Lucas Sequences

TL;DR

This paper extends $p$-adic valuation results for Lucas sequences to $p$-adic congruences, deriving new congruences and exploring prime entry point behavior, with conjectures on biases in Lucas sequences.

Contribution

It introduces new $p$-adic congruences for Lucas sequences and analyzes prime entry points, proposing a conjecture on Chebyshev-like biases.

Findings

Derived new $p$-adic congruences for Lucas sequences

Established constraints on prime entry points in Lucas sequences

Conjectured a Chebyshev-like bias in real Lucas sequences

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.