bibkey: ceradaconceicao2026rankdensity authors: Joaquim Cera Da Conceição year: 2026 title: “Explicit Prime Densities for Lucas Sequence Rank Divisibility” doi: null url: https://arxiv.org/abs/2604.20014v3 claim: “Closed-form densities for primes whose Lucas rank of appearance is divisible by a fixed integer d, completing general cases left by Sanna; the Fibonacci all-d density problem was already solved by Cubre and Rouse, and these densities are not pointwise bounds on divisors of a specified term.” strata_touched: [] license: citation-only triage: anchor

Lucas rank-divisibility density and specified-term divisibility

The primary source is arXiv:2604.20014v3, dated 22 August 2026; the first version appeared in April 2026. The locators below refer to the introduction, Theorem 1, and Section 2 of that version. They do not assert an independent proof audit or Lean verification.

The predicate and the published counting theorem

The paper takes nonzero integer parameters $a_1,a_2$ and defines $U_0=0$, $U_1=1$, $U_{n+2}=a_1{U}_{n+1}-a_2{U}_n$, and uses ${\rho }_U(p)=\min \{n\ge 1:p\mid U_n\}$ when it exists. For characteristic roots $a,b$, put $K=\mathbb{Q}(a)$, $\gamma =a/b$, and $\Delta =a_1^2-4a_2$. Section 2, printed p.4, assumes that $\gamma$ is not a root of unity and focuses on irreducible $X^2-a_{1}X+a_2$; the reducible case was settled by Wiertelak. These hypotheses remain in force for the counting statement below. Away from the displayed exceptional primes, the rank equals the order of $\gamma$ in the appropriate residue field.

For a fixed positive integer $d$, the counting function is

$$R_{\gamma }(d;x)=\#\{p\le x:p\nmid 2a_2\Delta ,d\mid {\mathrm{ord}}_{\mathfrak{p}}(\gamma )\text{ for every }\mathfrak{p}\mid p\}.$$

Theorem 1, printed pp.2–3, gives an absolute constant $B>0$ such that for $x>\exp (B{d}^{40})$,

$$R_{\gamma }(d;x)={\delta }_{\gamma }(d)\mathrm{Li}(x)+O_{\gamma }\left(\frac{d}{\phi (d)}\frac{x(\log \log x{)}^{\omega (d)}}{(\log x{)}^{9/8}}\right).$$

The source supplies a cyclotomic–Kummer degree series for ${\delta }_{\gamma }(d)$ and evaluates it in closed form in the remaining cases. The constant in the error term may depend on $\gamma$; it is not a uniform-in-sequence statement. The $d^{40}$ threshold must be retained when considering a varying $d$.

History prevents a duplicate Fibonacci result

The introduction, printed pp.1–2, attributes earlier cases to Hasse, Wiertelak, Lagarias, Ballot, and Sanna. It explicitly says that the Fibonacci case for all $d\ge 1$ was already completed by Cubre and Rouse, Divisibility properties of the Fibonacci entry point, Proceedings of the AMS 142 (2014), 3771–3785, DOI 10.1090/S0002-9939-2014-12269-6. Its original preprint is arXiv:1212.6221v1. Sanna’s general Lucas predecessor is On the divisibility of the rank of appearance of a Lucas sequence, IJNT 18 (2022), 2145–2156, DOI 10.1142/S1793042122501093.

The 2026 source completes missing general Lucas cases and finite-form evaluations. Its Theorem 1 also explains which asymptotic parts were already available from Sanna’s argument. Neither the Fibonacci density statement nor this general counting theorem should be reproved or presented as a new FIB-project result.

The order of divisibility is material

For Fibonacci, with $z(p)$ the first zero index, specified-term divisibility has the predicate $p\mid F_n⟺z(p)\mid n$. The density theorem instead counts $d\mid z(p)$ for fixed $d$. These conditions have opposite divisibility directions. A density for the infinite prime set in the latter does not by itself bound the finite set of actual prime divisors of one $F_n$, its reciprocal prime mass, or its divisor sum. A transport argument would have to preserve the actual condition and justify its moving parameters and errors.

For the general seed $V_j=a{F}_{j+3}+b{F}_{j+4}$, the FIB theory’s §191.1 already supplies the correct zero-set description: when nonempty, it is one residue class $j\equiv t_v(p)(\mathrm{mod}z(p))$. In general $t_v(p)$ is nonzero. The rank-density theorem does not locate that seed-dependent residue or prove that different primes’ residues are jointly attained by the same $j$.

The Kummer–Chebotarev construction in FIB §§199–201 already uses specified-root events for the actual source. Replacing those events by rank density would lose the target $v^{\prime }/v$ and its joint constraints. This note keeps the latest general rank result available for reuse, without claiming that it fills the unrestricted changing-seed estimate or the same-candidate divisor-weight budget in §233.5.