Consecutive Zeckendorf–Niven terms in a 2-AP: a two-parameter family of five-term runs, and six-term runs confined to two Fibonacci shapes

Abstract

A positive integer n is Zeckendorf–Niven when the number s_(Z)(n) of summands in its Zeckendorf representation divides n. Lao, Miller, Rosa, Shiliaev, Tresch, Wong and Zhang recently studied how many consecutive terms of an arithmetic progression of common difference two can be Zeckendorf–Niven. They proved that the only runs of eight or more terms are subsequences of 2,4,…,18, and constructed a five-term run starting at 27+F₁₂₀ⱼ₊₁₇ for every j; between five and seven the maximum outside that block is left undetermined. We prove three things. First, their construction is one line of a two-parameter family: for 8 ≤ a, a+2 ≤ b, 11 ≤ b and Fₐ+F_b ≡ 58 (mod 60), the number n=6+Fₐ+F_b starts a run of five, and the run is exactly five on both sides. Second, every n ≥ 21 that starts a run of six satisfies n+3=F_G or n+5=F_G for some G ≥ 21; beyond n ≥ 21 there is no hypothesis at all — none on the size of n, none on the position of its Zeckendorf digits. Third, neither of the two surviving one-parameter families produces a run of six for G ≤ 501, so no n with 21 ≤ n<F₅₀₀ ≈ 1.39 · 10¹⁰⁴ starts a run of six: throughout that range a run beginning outside 2,4,…,18 has at most five terms, and five occurs. What is left of the question is a single Fibonacci-index divisibility problem, stated in Question [q:residue]. All three results, and every lemma they rest on, are machine-checked in Lean 4.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   265a7cb57bb62f224dd7dd63e9ef2c9e626990bf41c0736e6a315ed3d2491e08