bibkey: browningverzobio2024sds authors: Tim Browning; Matteo Verzobio year: 2024 title: “Strong divisibility sequences and sieve methods” doi: null url: https://arxiv.org/abs/2402.19301v1 claim: “For strong divisibility sequences, sieve bounds control the density of indices whose terms have only large prime factors; the Fibonacci specialization is an index-density result and does not give a pointwise large-prime tail for every Fibonacci term.” strata_touched: [] license: citation-only triage: anchor

Strong divisibility sequences and sieve methods

The source is arXiv:2402.19301v1, by Tim Browning and Matteo Verzobio. It studies strong divisibility sequences (SDS), with Fibonacci and elliptic divisibility sequences as examples. The statements below are attributed to that version; no independent proof audit or Lean verification is claimed.

Theorem 1.2 gives a lower bound for the number of indices (n\le N) whose term has no prime divisor below

[ z=\tfrac12(\log N)(\log\log N), ]

under an eventual rank-of-apparition hypothesis (m_p<p^\alpha). Theorem 1.4 gives an upper bound for the same index set when almost every term has a primitive prime divisor. For Fibonacci numbers, Corollary 1.6 records the corresponding two-sided density scale. Theorem 1.5 shows, under its stated density condition, that prime terms in an SDS have density zero. Theorem 1.7 also gives a lower bound for the associated Eratosthenes–Legendre sieve product.

These are aggregate statements over indices. They do not imply the pointwise estimate displayed as a sufficient condition in FIB §195.4,

[ (\log j)\sum_{\substack{p>j^{5/6}\p\mid V_j}}\frac1{p-1}\longrightarrow0 ]

on a specified parity class. A sparse exceptional set of indices is compatible with the quoted density bounds.

The current FIB theory already supplies a different route beyond that older interface: §§199–201 contain paper conclusions for every fixed primitive seed and for explicit slowly growing norm families. Those results retain their stated analytic inputs and seed-dependent or restricted-uniform thresholds; they are not Lean verification. Section 195.4 must therefore not be described as an unresolved fixed-seed Robin theorem.

The remaining broad bridges include unrestricted changing seeds and the same actual candidate’s full divisor weight. In §233.5 the candidate is the unique remaining integer in the specified affine residue class, not automatically a product (gV_j). The sieve-density theorem supplies neither that joint weight nor a transport between the two families. It is retained as a historical and reusable sieve input, rather than as a reason to repeat the fixed-seed proofs.