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Cyclotomic Congruences and Lucas Sequences

Tyler Ross, Zhongyan Shen, Tianxin Cai

TL;DR

This work extends Carmichael’s $p$-adic valuation results to $p$-adic congruences for the Möbius duals of Lucas sequences, removing regularity constraints and yielding congruences for adjacent Lucas terms. It connects these congruences to prime-entry-point data by deriving corollaries for $M^U_n$, $M^V_n$ and ratios of Lucas terms, and by constraining the behavior of primes at entry points through a conjectured Chebyshev-like bias for real regular Lucas sequences. The analysis uses cyclotomic polynomials and local $p$-adic methods in quadratic fields, with concrete discussion of Fibonacci numbers and connections to Wall-Sun-Sun primes. Overall, the paper provides a detailed p-adic and cyclotomic framework for divisibility and distribution properties of Lucas sequences and proposes a new bias phenomenon in their prime-entry behavior.

Abstract

In this paper, we extend the $p$-adic valuations originally obtained by Carmichael for the sequences obtained by applying Möbius 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.

Cyclotomic Congruences and Lucas Sequences

TL;DR

This work extends Carmichael’s -adic valuation results to -adic congruences for the Möbius duals of Lucas sequences, removing regularity constraints and yielding congruences for adjacent Lucas terms. It connects these congruences to prime-entry-point data by deriving corollaries for , and ratios of Lucas terms, and by constraining the behavior of primes at entry points through a conjectured Chebyshev-like bias for real regular Lucas sequences. The analysis uses cyclotomic polynomials and local -adic methods in quadratic fields, with concrete discussion of Fibonacci numbers and connections to Wall-Sun-Sun primes. Overall, the paper provides a detailed p-adic and cyclotomic framework for divisibility and distribution properties of Lucas sequences and proposes a new bias phenomenon in their prime-entry behavior.

Abstract

In this paper, we extend the -adic valuations originally obtained by Carmichael for the sequences obtained by applying Möbius 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.

Paper Structure

This paper contains 5 sections, 10 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. A Conjecture
  4. Auxiliary Results
  5. Proofs of Main Results

Key Result

Theorem 2.1

Fix any positive integers $p, n \geq 1$, with $p$ prime and $(p,n)=1$. If $p \nmid (P,Q)$, we have the following congruences. If $p \mid (P, Q)$, then instead we have the following congruences.

Figures (1)

  • Figure 1: Bias in Fibonacci entry points. Here, $\# Z^R_F(n)$ is the solid line, $\# Z^N_F(n)$ the dotted line.

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Conjecture 3.1
  • Proposition 4.1: Doubling formula, Ross
  • Proposition 4.2: Ross
  • Lemma 4.3: Leb, Lehm, Wash
  • ...and 7 more