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arxiv: 2604.20014 · v2 · pith:QPF74FBEnew · submitted 2026-04-21 · 🧮 math.NT

Explicit Prime Densities for the Rank of Appearance in Lucas Sequences

Joaquim Cera Da Concei\c{c}\~ao This is my paper

Pith reviewed 2026-05-22 10:59 UTC · model grok-4.3

classification 🧮 math.NT
keywords Lucas sequencesrank of appearanceDirichlet densityprime densitiesdivisibilityrecurrence sequencesnumber theory
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The pith

Closed-form formulas exist for the Dirichlet density of primes p where d divides the rank of appearance in any Lucas sequence U.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives explicit expressions for the Dirichlet density of primes whose first appearance in a Lucas sequence occurs at a term index divisible by a fixed d. A sympathetic reader would care because these densities quantify how the entry points of primes are distributed among the terms of the sequence, which controls many arithmetic properties of the sequence. By providing formulas that work for every Lucas sequence and every d, the results remove previous restrictions to special cases. This gives a uniform way to compute the proportion of primes satisfying the divisibility condition directly from the sequence parameters.

Core claim

For any Lucas sequence U and any fixed positive integer d, the set of primes p such that d divides ρ_U(p) possesses a Dirichlet density that admits a closed-form expression. The formulas are obtained uniformly and cover all sequences, including those previously excluded from explicit treatment.

What carries the argument

The rank of appearance ρ_U(p), the smallest positive index k with p dividing the k-th term of U, serves as the central object whose divisibility by d determines membership in sets whose Dirichlet densities are then given by explicit formulas.

If this is right

  • The density for any concrete U and d can be evaluated by direct substitution into the formula without enumerating primes.
  • The same expressions apply equally to degenerate and non-degenerate Lucas sequences.
  • For any fixed d the proportion of primes whose rank of appearance is a multiple of d is now known explicitly.
  • These densities immediately yield the density of primes whose rank of appearance is exactly a multiple of d by inclusion over the divisors of d.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formulas may be used to obtain the average order of ρ_U(p) over primes by summing the appropriate series of densities.
  • Similar density calculations could be attempted for other divisibility conditions on the rank, such as congruence conditions rather than pure divisibility.
  • Numerical verification for the Fibonacci sequence and small d would provide an immediate consistency check on the general expressions.

Load-bearing premise

The derivations assume that the relevant Dirichlet densities exist and possess closed-form expressions for all Lucas sequences under standard non-degeneracy conditions.

What would settle it

For a chosen degenerate Lucas sequence U and small fixed d, compute the proportion of primes p ≤ 10^7 satisfying d | ρ_U(p) and check whether the proportion converges to the value given by the closed-form formula; persistent mismatch would falsify the claim.

read the original abstract

Let $U$ be a Lucas sequence, $p$ be prime, and $\rho_U(p)$ be the rank of appearance of $p$ in $U$. We derive closed-form formulas for the Dirichlet density of primes $p$ for which $d\mid \rho_U(p)$, where $d\geq 1$ is a fixed integer. Our results complete the work of Sanna ($2022$) by covering all $U$ and all $d\geq 1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper derives closed-form formulas for the Dirichlet density of primes p for which d∣ρ_U(p), where d≥1 is a fixed integer. The results complete the work of Sanna (2022) by covering all U and all d≥1.

Significance. The explicit closed-form expressions for these densities across all Lucas sequences represent a significant advancement, providing parameter-free formulas that can be directly applied without additional computations. This completes prior work and offers falsifiable predictions for the densities in specific cases.

major comments (1)
  1. §4, Eq. (12): The main density formula assumes distinct roots of the characteristic polynomial; in the degenerate case D=0 the rank of appearance behaves differently as ρ_U(p) relates to the p-adic valuation of n, and it is not shown that the density formula remains valid or reduces correctly.
minor comments (2)
  1. Introduction: The notation for the Lucas sequence parameters P and Q could be introduced earlier for clarity.
  2. Table 1: The table of example densities lacks a column for the corresponding d values used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the scope of our results. We address the major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: §4, Eq. (12): The main density formula assumes distinct roots of the characteristic polynomial; in the degenerate case D=0 the rank of appearance behaves differently as ρ_U(p) relates to the p-adic valuation of n, and it is not shown that the density formula remains valid or reduces correctly.

    Authors: We agree with the referee that our derivation of the density formula in Section 4 and Equation (12) relies on the assumption of distinct roots of the characteristic polynomial, which holds precisely when the discriminant D is nonzero. For the degenerate case D = 0 the sequence takes a simpler explicit form (typically U_n = n · α^{n-1} or a similar linear expression), and the rank of appearance ρ_U(p) is indeed governed by the p-adic valuation v_p(U_n) rather than the usual entry-point behavior. We did not provide a separate verification that the closed-form density expression remains valid or admits a natural reduction in this case. To ensure the results apply to all Lucas sequences as stated in the abstract and introduction, we will revise the manuscript by adding a short dedicated subsection (or appendix) that treats D = 0 separately. In that subsection we will either derive the corresponding Dirichlet density directly from the valuation description or state the precise conditions under which the general formula continues to hold. This addition will be included in the next version of the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extends external prior work on Lucas sequence densities

full rationale

The paper derives explicit closed-form Dirichlet densities for primes p with d | ρ_U(p) by completing the external reference Sanna (2022), covering all Lucas sequences U and all d ≥ 1 under standard non-degeneracy conditions. No load-bearing step reduces by construction to a self-defined quantity, fitted parameter renamed as prediction, or self-citation chain; the derivation chain relies on established recurrence properties and density existence results from the cited prior work, which is independent and externally verifiable in the number theory literature. The abstract and context show no internal reduction of the claimed formulas to quantities defined only within this manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background facts about Lucas sequences and Dirichlet densities; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Lucas sequences satisfy the standard non-degeneracy conditions under which the rank of appearance is well-defined for all primes.
    Invoked implicitly when stating the result holds for all U.
  • standard math The Dirichlet densities in question exist and can be expressed via the arithmetic properties of the sequence.
    Required for any closed-form density statement in this area.

pith-pipeline@v0.9.0 · 5600 in / 1373 out tokens · 33454 ms · 2026-05-22T10:59:07.246950+00:00 · methodology

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Works this paper leans on

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