bibkey: lucatron2015selffibonacci authors: “Florian Luca; Emanuele Tron” year: 2015 title: “The Distribution of Self-Fibonacci Divisors” doi: 10.1007/978-1-4939-3201-6_6 url: https://arxiv.org/abs/1410.2489v2 claim: “Theorem 2.2 uses strict descent of prime factors of Fibonacci entry points, and Section 4 explicitly describes Pratt-Fibonacci trees built from factors of z(p).” strata_touched: [] license: citation-only triage: anchor

Fibonacci rank descent and Pratt-Fibonacci trees

The primary text is arXiv:1410.2489v2. The published chapter is in Advances in the Theory of Numbers, Fields Institute Communications 77, pages 149–158, DOI. The locators below use the eight-page arXiv version.

For the standard Fibonacci sequence, write (z(n)=\min{j\ge1:n\mid F_j}). The proof of Theorem 2.2 on page 4 uses the fact that, for a prime (P\ge7), every prime factor of (z(P)) is strictly smaller than (P). It applies this descent, together with prime-power valuation information, to iteration of (z) on integers.

Section 4, pages 7–8, explicitly calls the corresponding objects “Pratt-Fibonacci trees”: the factors of (z(p)), with their exponents, appear as children of a node (p^\delta). Thus the prime-factor dependency construction and the descent mechanism are literature-attested; renaming them does not give a new construction.

Observer and Arithmetic Relations §24 identifies nodes with the same prime value, absorbs (2,3,5), and bounds the resulting directed-acyclic-graph height by (\lfloor\log_2(p-1)\rfloor). Its height argument is supplied in that theory entry. The cited paper is the source for the dependency relation and descent, not a citation for that exact height statement or a Lean verification of the consumer.

Lemma 3.1 on page 6 quotes Gordon–Pomerance’s uniform estimate for integers with a prescribed rank. Its (x^{1-o(1)}) scale and its integer counting domain do not supply the fixed-support prime count in §26 of the observer volume.

Verified locator

• DOI: https://doi.org/10.1007/978-1-4939-3201-6_6
• URL: https://arxiv.org/abs/1410.2489v2
• Version and location: arXiv:1410.2489v2, revised 20 February 2015; the arXiv record lists the DOI above as the related publication.