bibkey: fiebigmbirikaspilker2025lucas authors: Morgan Fiebig, aBa Mbirika, Jürgen Spilker year: 2025 title: “Period patterns, entry points, and orders in the Lucas sequences: theory and applications” doi: null url: https://arxiv.org/abs/2408.14632v2 claim: ‘Conjecture 5.2. Set $U_n:=U_n(p,q)$, and let $m\ge 1$ and $n\in \mathbb{Z}$. Assume that both $p$ and $m$ are even. If $U_{2n}\equiv 0(\mathrm{mod}m)$ and $U_{2n+1}\equiv q^n(\mathrm{mod}m)$, then exactly one of the following two conclusions occur: (i) $U_n\equiv 0(\mathrm{mod}m)$, or (ii) $U_n\equiv \frac{m}{2}(\mathrm{mod}m)$ where $n=c\cdot \frac{e_U(m)}{2}$, where $c$ is some odd integer and $e_U(m)$ is the entry point of $m$ in $(U_n{)}_{n\ge 0}$.’ strata_touched:

• D5/S1/Recurrence/LucasEvenDescent license: citation-only triage: anchor

Period patterns, entry points, and orders in the Lucas sequences

The paper extends the modular theory of the Fibonacci and Lucas sequences to the two-parameter Lucas sequences U_n(p,q) and V_n(p,q), where U_0 = 0, U_1 = 1, V_0 = 2, V_1 = p, and both satisfy x_{n+1} = p*x_n - q*x_{n-1}. Its statistics are the period pi(m), the entry point e(m), and the order omega(m) = pi(m)/e(m).

Conjecture 5.2, quoted above, is the paper’s own open problem, carried in Section 5 (“Open questions and future work”). The paper attributes it to observations by Diego Garcia-Fernandezsesma and Oliver Lippard together with Mbirika, following the Problem Session of the 21st International Fibonacci Conference; the index form in clause (ii) is credited to Lippard. It repairs a statement whose sufficiency direction the paper shows holds only when p or m is odd, and asks what happens when both are even.

Three conventions of the paper are inherited by the conjecture and are quoted here because the third is load-bearing: the parameters satisfy gcd(p,q) = 1 with p and q nonzero; the sequences are nondegenerate, so q != 0 and alpha/beta is not a root of unity; and the moduli are restricted to those with gcd(q,m) = 1, “for otherwise, the sequence may not be purely periodic according to our definition”, which is also what guarantees that e_U(m) exists. The entry point is defined there as “the least integer r>0 (if it exists) such that m divides S_r”.

The module proves the conjecture. It does not use the first two conventions; the implication holds without any nonzero, coprimality or nondegeneracy restriction on p and q, so the formal statement is more general than the conjecture in those parameters. It does use gcd(q,m) = 1.

Verified locator

• Abstract, version history and journal reference: https://arxiv.org/abs/2408.14632v2
• The conjecture and the three conventions were read in the LaTeX source obtained from https://arxiv.org/e-print/2408.14632v2 (Conjecture 5.2 at the label conj:aBa_Diego_Oliver; the modulus convention at the label conv:gcd_of_m_and_q_equals_1).
• Journal reference as listed on that page: The Fibonacci Quarterly 63.2 (2025) 345-376.