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Consecutive Divisors of Sums of Fibonacci Polynomials: a Luca–Marques Analogue in Finite-Field Polynomial Rings

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Scott Kominers ↗

Version 1 / Nov 10, 2026 / CC BY-SA 4.0

Abstract

For every prime power qq with gcd⁡(q,6)=1\gcd(q,6)=1, we construct infinitely many pairs of consecutive monic polynomials in Fq[T]\mathbb{F}_q[T] whose product divides the Fibonacci-polynomial sum with upper index one less than their common norm. This gives a function-field counterpart of the divisibility condition in the Luca–Marques theorem on consecutive integral arithmetic means of Fibonacci numbers. A short extension of the construction gives infinitely many triples of consecutive monic polynomials with the same property.

Author comments: Dedicated to the memory of Florian Luca

Provenance statement

I used LLMs to assist with computations and analysis in the preparation of this article, particularly GPT-5.5 Pro and Claude 4.7 Opus (accessed in part via Poe with the support of Quora, where I am an advisor). I thank the referee for a careful reading and for suggesting the question that led to Corollary 8.1.

Tools used

OpenAI
ChatGPTVersion 5.5 Pro
Anthropic
Claude OpusVersion 4.7
Quora
Poe

References

  1. Amirali Fatehizadeh and Daniel Yaqubi, Average of the Fibonacci numbers , J. Integer Seq. 25 (2022), no. 2, Article 22.2.6. https://cs.uwaterloo.ca/journals/JIS/VOL25/Yaqubi/yaq6.html.link
  2. Oisín Flynn-Connolly, On the divisibility of sums of Fibonacci numbers , Integers 25 (2025), Article A87. https://math.colgate.edu/~integers/z87/z87.pdf. Errata (2026): https://math.colgate.edu/~integers/z87/z87.errata.pdf.link
  3. Florian Luca and Diego Marques, On integral consecutive arithmetic means of the first Fibonacci numbers , Arch. Math. 125 (2025), no. 5, 505--511. https://doi.org/10.1007/s00013-025-02166-2.DOI

Version history

  1. v1 · 8 pagesSubmitted by Scott KominersDedicated to the memory of Florian LucaCurrentNov 08, 2026