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Explicit Prime Densities for Lucas Sequence Rank Divisibility

This paper derives closed-form formulas for the Dirichlet density of primes $p$ such that a fixed integer $d$ divides the rank of appearance of $p$ in a Lucas sequence $U$ , thereby completing the work of Sanna (2022) by covering all Lucas sequences and all integers $d \geq 1$ .

Original authors: Joaquim Cera Da Conceição

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Joaquim Cera Da Conceição

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of number theory, mathematicians often study sequences of numbers that grow according to a simple, repeating rule. Imagine a line of numbers where each new entry is created by combining the two that came before it, much like how a family tree branches out from a common ancestor. These are known as Lucas sequences. A central question for researchers is to understand how these sequences interact with prime numbers—the indivisible building blocks of arithmetic. For any given prime number, there is a specific moment in the sequence where that prime first appears as a factor. Mathematicians call this the "rank of appearance." It is a unique fingerprint for that prime within that specific sequence. The big question is: if we look at all the prime numbers, how frequently do we find ones whose rank of appearance is divisible by a specific number, say three, or five, or a hundred? This frequency is known as the Dirichlet density, a way of measuring how common a certain pattern is among the infinite set of primes.

For decades, mathematicians have been able to calculate this frequency for many specific cases, but a complete picture remained elusive. Previous work had solved the problem for certain types of sequences and specific divisors, but gaps remained, particularly when the underlying mathematical structure involved complex symmetries related to the number three or the square root of negative one. These gaps were like missing pieces in a puzzle that prevented a full understanding of the landscape. The paper at hand steps in to fill those final holes. The author has derived exact, closed-form formulas that allow anyone to calculate the precise density of primes for which the rank of appearance is divisible by any chosen integer, regardless of the specific Lucas sequence being studied. This work completes a long-standing effort, ensuring that the rules governing these patterns are now known for every possible scenario.

The researchers approached this by translating the problem of finding prime factors into a problem about the behavior of numbers in extended mathematical worlds. Instead of looking at the sequence directly, they examined how a specific ratio of the sequence's roots behaves when reduced modulo a prime. This ratio acts like a generator, and its order in a specific group determines the rank of appearance. The challenge was that the behavior of this generator changes depending on the properties of the prime and the sequence. The author had to account for various scenarios, such as when the sequence's defining polynomial splits in a field containing the square root of negative three, or when it involves the square root of negative one. In these special cases, the standard methods of calculation failed because the usual symmetries were disrupted by the presence of these specific roots.

To overcome this, the team developed a systematic method to handle these difficult cases. They broke the problem down into smaller, manageable parts, analyzing the structure of the mathematical fields where these roots live. They determined exactly when certain automorphisms—transformations that preserve the structure of these fields—exist. These transformations are crucial because they dictate how the primes are distributed. By carefully mapping out these conditions, the author was able to write down a single, unified formula that works for all integers and all Lucas sequences. This formula is "closed-form," meaning it can be computed using a finite number of standard operations, rather than requiring an infinite process or a guess-and-check approach.

The paper confirms that the density of these primes is not just a theoretical possibility but a concrete, calculable value. The author did not merely suggest that these values exist; they proved their existence and provided the exact recipe to find them. They also addressed a subtle issue regarding the "rank" when the divisor is even, showing how to adjust the calculation to account for the specific parity of the numbers involved. Their results build upon and refine earlier work by other mathematicians, removing arbitrary restrictions that had previously limited the scope of the solution. For instance, earlier studies had to exclude cases where the divisor was a multiple of two or three in certain fields, but this new work removes those exclusions entirely.

To ensure their formulas were correct, the researchers tested them against real data. They wrote computer programs to generate millions of prime numbers and calculated the actual frequency of the pattern in question. They then compared these experimental results with the values predicted by their new formulas. The match was precise, with the calculated values aligning with the computer-generated data to six decimal places. This verification step was vital, as it confirmed that the abstract formulas accurately reflected the reality of the number system. The paper includes tables showing these comparisons for various sequences and divisors, demonstrating that the theory holds up under rigorous testing.

Ultimately, this work provides a complete map for a specific corner of number theory. It tells us exactly how often a prime number will appear at a position in a Lucas sequence that is a multiple of a given number. Whether the sequence is the famous Fibonacci sequence or a more obscure variation, and whether the divisor is a small number like two or a large, complex integer, the answer is now known. The author has closed the book on this particular problem, leaving no loose ends and no unexplored cases. Their contribution is a testament to the power of persistence in mathematics, turning a fragmented collection of partial answers into a single, coherent, and complete theory.

Technical Summary: Explicit Prime Densities for Lucas Sequence Rank Divisibility

Problem Statement
This paper addresses the problem of determining the Dirichlet density of the set of primes $p$ for which the rank of appearance $\rho_U(p)$ in a Lucas sequence $U = (U_n)_{n \ge 0}$ is divisible by a fixed integer $d \ge 1$ . The rank of appearance is defined as the least positive integer $n$ such that $p \mid U_n$ . While the existence of this density was established in prior work, explicit closed-form formulas were incomplete, particularly for cases involving cyclotomic splitting fields and specific divisibility conditions on $d$ (specifically when $d$ is divisible by 2 or 3).

Context and Previous Work
The study builds upon foundational work by Hasse, who treated specific square-free parameters and prime $d$ , and Wiertelak, who solved the case where the characteristic polynomial is reducible. Significant progress was made by Lagarias, Cubre, Rouse, and Ballot for specific sequences and small values of $d$ . Most recently, Sanna (2022) provided a general framework and explicit formulas for odd $d$ not divisible by 3, provided the splitting field of the characteristic polynomial is not $\mathbb{Q}(\sqrt{-3})$ . However, Sanna's results left open the cases where the splitting field is cyclotomic (specifically $\mathbb{Q}(i)$ or $\mathbb{Q}(\sqrt{-3})$ ) and cases where $d$ is divisible by 2 or 3 in these fields.

Methodology
The author approaches the problem by translating the divisibility of the rank $\rho_U(p)$ into a condition on the multiplicative order of the root quotient $\gamma = a/b$ modulo prime ideals in the quadratic field $K = \mathbb{Q}(a)$ . Specifically, $\rho_U(p)$ is divisible by $d$ if and only if the order of $\gamma$ modulo $\pi$ is divisible by $d$ .

The core of the methodology involves the study of Cyclotomic-Kummer extensions of the form $K_{n,d} = K(\zeta_n, \gamma^{1/d})$ , where $\zeta_n$ is a primitive $n$ -th root of unity. The Dirichlet density is expressed as a sum over divisors of $d$ involving the degrees of these field extensions and the existence of specific automorphisms in their Galois groups.

Key technical steps include:

Field Degree Computation: Utilizing Theorem 2 and Theorem 3 (from standard literature) to determine the degree $[K_{n,d} : \mathbb{Q}]$ . This requires analyzing whether $\gamma$ is a power in cyclotomic extensions, which depends on the discriminant $\Delta_K$ and the norm of $\gamma$ .
Automorphism Existence: Determining the existence of automorphisms $\sigma_{u,v}$ in the Galois group that satisfy specific conditions (mapping $\sqrt{\Delta} \to -\sqrt{\Delta}$ , $\zeta \to \zeta^{-1}$ , and $\gamma^{1/uv} \to \gamma^{-1/uv}$ ).
Case Analysis based on $h$ : The author defines $h = \max_{\zeta \in \mu(K)} h(\zeta)$ $h = max_{ζ \in μ (K)} h (ζ)$ , where $h(\zeta)$ $h (ζ)$ is the largest integer such that $\zeta\gamma$ $ζ γ$ is an $h(\zeta)$ $h (ζ)$ -th power in $K$ $K$ . The paper is divided into two main analytical phases:
- Case $h = h(1)$ : The maximum is achieved at $\zeta = 1$ . The author derives closed-form formulas by decomposing the density into sums $S_{d,e,h}(\nu)$ and evaluating them using Lemma 11. This section handles the discriminants $\Delta_K \notin \{-4, -3\}$ , $\Delta_K = -4$ , and $\Delta_K = -3$ separately, utilizing Lemmas 4, 5, and 6 to handle square and cube roots in cyclotomic fields.
- Case $h \neq h(1)$ : The maximum is achieved at a root of unity $\zeta \neq 1$ . The author employs a switching formula (Theorem 20) to relate the density of $\gamma$ to that of $-\gamma$ (or $\zeta\gamma$ ), effectively reducing these cases to the $h=h(1)$ scenario or previously solved cases.

Key Contributions and Results
The primary contribution of this paper is the derivation of closed-form formulas for the Dirichlet density $\delta_\gamma(d)$ for all Lucas sequences $U$ and all integers $d \ge 1$ . This completes the work initiated by Sanna (2022).

Specific results include:

General Formulas: The density is expressed as a linear combination of terms involving the Möbius function, Euler totient function, and specific arithmetic conditions on the discriminants of auxiliary fields ( $K_1, K_2$ ) and conductors ( $f(L)$ ).
Resolution of Cyclotomic Cases: The paper explicitly resolves the difficult cases where the splitting field is $\mathbb{Q}(i)$ ( $\Delta_K = -4$ ) or $\mathbb{Q}(\sqrt{-3})$ ( $\Delta_K = -3$ ), including scenarios where $d$ is even or divisible by 3.
Handling of $h \neq h(1)$ : Theorem 20 provides a mechanism to handle cases where the maximal power condition is met by a non-trivial root of unity, ensuring the solution is exhaustive.
Numerical Verification: Section 6 provides numerical data comparing the derived closed-form values with experimental densities computed via SageMath for various sequences and values of $d$ , showing agreement up to 6 decimal places for primes up to $10^7$ .

Significance
The paper claims to provide a complete solution to the problem of computing the Dirichlet density of primes with rank divisible by $d$ in Lucas sequences. By covering all cases left by Sanna's theorem, including the intricate cyclotomic splitting fields and all divisibility conditions on $d$ , the author offers a comprehensive, closed-form characterization of these densities. The work relies on the existence of the density (proven by Sanna) and focuses on the explicit computation, thereby "completing" the theoretical landscape for this specific number-theoretic problem. The results are presented as final formulas requiring only finitely many operations to compute, avoiding asymptotic approximations for the density value itself.

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