Divisible Sequences

• Divisible sequences are defined by the property that m divides n implies aₘ divides aₙ, forming a cornerstone in number theory and algebraic dynamics.
• They encompass classical examples like Fibonacci, Lucas, and Mersenne sequences, and extend to elliptic, matrix, and non-classical constructions.
• Recent research has classified linear divisible sequences and unveiled connections to cyclotomic factorizations, Galois theory, and advanced sieve methods.

A divisible sequence is an integer or algebraic sequence possessing a robust divisibility structure: for all positive integers $m$ and $n$, $m \mid n \implies a_m \mid a_n$. This property underpins a deep arithmetic theory, influencing areas from elliptic and algebraic dynamics to analytic number theory, and finds systematic generalization in strong divisibility, linear divisibility, matrix-valued sequences, and higher-genus recurrences. Modern research has fully classified the linear case, elucidated the algebraic and analytic structure of prime divisors, and established connections to Galois theory, cyclotomic factorizations, and sieve theory.

1. Fundamental Definitions and Classical Structure

A sequence $\{a_n\}_{n\ge1}$ in a commutative ring $R$ is a divisibility sequence if $m \mid n$ implies $a_m \mid a_n$ in $R$ (Billal, 2015, Granville, 2022). Stronger forms impose:

• Strong divisibility: $\gcd(a_m, a_n) = a_{\gcd(m,n)}$ for all $m,n \ge 1$ (Billal, 2015, Browning et al., 2024).
• Linear divisibility sequence (LDS): In addition to divisibility, $\{a_n\}$ satisfies a homogeneous linear recurrence with constant coefficients (Abrate et al., 2017, Granville, 2022).
• Binomid sequences: Sequences $f_n$ for which all generalized binomial coefficients built from $f_n$ are integral, with a divisor-product structure ensuring integrality and divisibility at all "pyramid levels" (Shapiro, 2023).

Key classical examples include the Fibonacci, Lucas, and Mersenne sequences, all of which exhibit both divisibility and strong divisibility.

2. Structural Theorems and Classification

Recent work has yielded a complete structural classification of linear divisibility sequences in $\mathbb{Z}$ and over polynomial rings (Granville, 2022, Abrate et al., 2017, Koshkin, 2022). Every integer LDS decomposes uniquely into a finite product (up to periodic sign):

$$u_n = \varepsilon_n \cdot d^{-g_d} \cdot (n/d)^{e} \cdot T^n \cdot \prod_{j=1}^r \frac{f_j(\alpha_j^n)}{f_j(\alpha_j)}$$

where $d = \gcd(n,M)$, $\varepsilon_n$ is periodic, $e$ is an exponent, $T$ is an integer base for exponential blocks, and $f_j(X)$ are least common multiples of cyclotomic-type polynomials evaluated at algebraic numbers $\alpha_j$. This structure theorem extends verbatim to polynomial LDS (Granville, 2022).

A related GCD property holds: For all $m, n$, $\gcd(u_m, u_n) = \pm u_{\gcd(m, n)}$, provided the LDS is non-degenerate and normalized, unifying phenomena observed in Fibonacci and Lucas sequences.

For higher-order recurrences (e.g., order-4), the characteristic polynomials of non-degenerate LDS must possess a prescribed symmetric structure arising from Kronecker products, and factorization into order-2 LDS is generically possible (Abrate et al., 2017).

3. Strong Divisibility, Laws of Apparition, and LCM Sequences

Strong divisibility sequences are characterized by unique features such as:

• Law of Apparition: For each prime $p$, there is a minimal index $\rho(p)$ so that $p|\ a_n$ iff $\rho(p) | n$ (Billal, 2015). There are analogues for prime powers.
• LCM-Sequence Structure: Every SDS admits a unique "lcm-sequence" $\{b_n\}$ such that $a_n = \prod_{d|n} b_d$ and $\gcd(b_m, b_n) = 1$ if $m \nmid n$ and $n \nmid m$.
• Law of Repetition: For certain strong divisibility sequences, the $p$-adic valuation satisfies $\nu_p(a_{nk}) = \nu_p(a_n) + \nu_p(k)$ for $p \nmid k$, giving rise to precise recurrence and divisibility propagation (Billal, 2015).

Linear SDS examples, including the classical Fibonacci and Lucas sequences, saturate these laws. The set of terms that are prime in a given SDS has density zero, fundamentally shaped by the divisor-closed structure of $\{n: a_n=1\}$ (Browning et al., 2024).

4. Extensions: Matrix, Elliptic, and Nonclassical Sequences

Matrix Divisibility Sequences

Defining a sequence of $d \times d$ matrices $\{M_n\}$ with $n \mid m \implies \exists Q : M_m = Q M_n$, the determinant $\det(M_n)$ is a classical divisibility sequence (Górnisiewicz, 2015, Cornelissen et al., 2011). Determinant divisibility sequences from powers of integer matrices generalize Lucas sequences and encode higher-order factorization phenomena.

Elliptic Divisibility Sequences (EDS)

Given an elliptic curve $E/\mathbb{Q}$ and a non-torsion point $P$, the EDS $\{D_n\}$ is derived from the denominator of the $x$-coordinate of $[n]P$. EDS are strong divisibility sequences exhibiting quadratic exponential growth, nontrivial rank of apparition, and rich divisor structures. The set $S(D):=\{n: n\mid D_n\}$ admits a directed-graph structure: new elements in $S(D)$ are constructed via multiplication by divisors of $D_n$ and (coprime) aliquot products defined by cycles in the EDS's rank-of-apparition structure (Stange et al., 2010). The EDS context recovers and extends Lucas-Fibonacci results, provides analogues for prime-divisor propagation, and underlies many open problems in Diophantine and arithmetic dynamics.

Non-Classical Constructions

By utilizing polynomials with cyclotomic factors and combinatorial data encoded in labeled Hasse diagrams, non-classical linear divisibility sequences can be constructed which lack the strong divisibility property, i.e., for which $\gcd(a_m, a_n) \ne a_{\gcd(m,n)}$ in general (Koshkin, 2022). These are fully classified via the factorization structure of their defining polynomials, with new integer sequences constructed by evaluating these polynomials on algebraic integers.

Arithmetic and Integrability Properties

The Binomid Pyramid framework organizes sequences according to the integrality of their generalized binomial coefficients. For divisor-product sequences, full integrality at every level is guaranteed, and for Lucas sequences (and their divisor-product relatives), these diagrams clarify the embedding of the classical combinatorics into more general arithmetic settings (Shapiro, 2023).

5. Prime Divisor Density, Sieve Bounds, and Galois Theory

Analytic methods have quantified the density and distribution of prime divisors in SDS and related sequences:

• Upper bounds on the density of primes dividing a given strong or polynomially generated divisibility sequence do not exceed $1/2$; numerical experiments for second-order sequences yield typical densities closer to $0.35$ (Lipinski et al., 2021).
• Sieve methods yield lower bounds for the density of terms with all prime divisors exceeding a (slowly growing) threshold $z$, establishing that a positive proportion of terms are "z-rough," but the set of prime terms is sparse (Browning et al., 2024).
• The Galois structure (e.g., the reducibility of terms over function fields) and Chebotarev-type arguments underlie results on irreducibility and on the structure of primitive divisors in both Lucas and EDS contexts, with deep implications for arithmetic dynamics and diophantine undecidability (Ingram et al., 2011).

6. Generalizations and Open Problems

Divisibility sequences are further generalized to:

• Higher-order recurrences: Complete structure theorems are known in order 4 (factorization through Kronecker products of degree-2 LDS and construction from Salem numbers) (Abrate et al., 2017).
• Arithmetic divisibility sequences: Somos-4 and -5 sequences are shown to be arithmetic divisibility sequences, with prime-divisibility sets forming arithmetic progressions due to underlying elliptic divisibility companion recurrences (Kamp, 2015).
• Function fields and rings: The complete characterization extends to function fields and homogeneous polynomial rings; these results interact with the theory of algebraic divisibility sequences and amplify the range of possible base fields.

Open areas include the precise classification of strong divisibility in higher-order recurrences; a full asymptotic theory of prime appearance; explicit “index divisibility” descriptions in elliptic and higher-order contexts; and the analytic behavior of divisor-closed sets in sparse arithmetic sequences.

7. Representative Examples

Sequence Type	Example Formula	Divisibility Law
Fibonacci sequence	$F_n = F_{n-1} + F_{n-2}$	$m \mid n \implies F_m \mid F_n$
Lucas sequence	$U_n = (\alpha^n - \beta^n)/(\alpha-\beta)$	$m|n \implies U_m|U_n$
EDS (elliptic, $D_n$)	$x([n]P) = A_n/D_n^2$	$m|n \implies D_m|D_n$
Determinant divisibility seq.	$$D_n = n^2 d^{n-1} ((\alpha^n - \beta^n)/(\alpha-\beta))^2$$	$n|m \implies D_n|D_m$
Mersenne numbers	$2^n - 1$	$m|n \implies 2^m-1|2^n-1$

These structured classes exemplify the general theory and embody the profound interplay between recurrence, divisibility, algebraic structure, and analytic density.

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References:

• "Classifying linear division sequences" (Granville, 2022)
• "Strong divisibility sequences and sieve methods" (Browning et al., 2024)
• "Divisibility Properties for Integer Sequences" (Shapiro, 2023)
• "Memoir on Divisibility Sequences" (Billal, 2015)
• "Matrix divisibility sequences" (Cornelissen et al., 2011)
• "New examples of determinant divisibility sequences" (Górnisiewicz, 2015)
• "Terms in elliptic divisibility sequences divisible by their indices" (Stange et al., 2010)
• "Linear divisibility sequences and Salem numbers" (Abrate et al., 2017)
• "Non-classical linear divisibility sequences and cyclotomic polynomials" (Koshkin, 2022)
• "Subsequences and Divisibility by Powers of the Fibonacci Numbers" (Onphaeng et al., 2013)
• "Somos-4 and Somos-5 are arithmetic divisibility sequences" (Kamp, 2015)
• "On Sequence Groups" (Lipinski et al., 2021)
• "Algebraic divisibility sequences over function fields" (Ingram et al., 2011)