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A Fibonacci Analogue of the Fifteen Theorem

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

Contributed by Scott Kominers ↗

Version 1 / Nov 12, 2026 / CC BY-SA 4.0

Abstract

We prove that a positive-definite quadratic form with integral Gram matrix represents every Fibonacci number if and only if it represents the first forty-two of them. The unique minimal criterion set consists of ten Fibonacci numbers, the largest of which is F42=267914296F_{42}=267914296.

Provenance statement

I used LLMs to assist with coding, computations, analysis, and revision, including GPT-6 Astra, GPT-5.2 Pro, GPT-5.5 Pro, GPT-5.6 Sol, and Claude Fable 5 (all accessed in part via Poe with the support of Quora, where I am an advisor). The Lean formalization was developed using Claude Opus 5.5, Claude Fable 5.2, and GPT-6 Astra. GPT-6 Astra identified a key step that made it possible to make my earlier conditional argument unconditional.

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Version history

  1. v1 · 37 pagesSubmitted by Scott KominersInitial depositCurrentNov 11, 2026