bibkey: leonetti2017fibonaccigcd authors: Paolo Leonetti and Carlo Sanna year: 2017 title: On the greatest common divisor of n and the nth Fibonacci number doi: null url: https://arxiv.org/pdf/1704.00151v2 claim: Lemma 2.4 bounds a weighted Fibonacci entry-point tail; Lemma 2.2(v) identifies the prime lcm outside the exceptional prime five. strata_touched: [] license: citation-only triage: anchor

A published weighted entry-point tail

The inspected source is arXiv:1704.00151v2, pages 3–4. Write z(p) for the Fibonacci entry point and ell(p) = lcm(p,z(p)). Lemma 2.4 states

$$\sum _{p>y}\frac{1}{\phi (\ell (p))}\ll y^{-1/4},y>0.$$

Lemma 2.2(v) gives ell(p) = p z(p) for primes other than five. Since phi(k) <= k, these two results imply

$$\sum _{p>y}\frac{1}{pz(p)}\ll y^{-1/4},y\ge 5.$$

This is a sufficient published input for the density-one argument in FIB §192. At cutoff y = j^(5/6), it gives an exceptional-index count O_v(X^(19/24) (log X)^2) after the stated averaging and threshold argument. The fixed seed and the allowed multiplier constant remain fixed throughout that limit. The argument does not remove all exceptional indices.

FIB §191 separately records the elementary bucket estimate sum_(p>y) 1/(p z(p)) <= 16/sqrt(y). Its resulting exponent 7/12, the uniform variable-multiplier Robin estimate and the canonical-source interpretation are project paper deductions, not claims attributed to this article. No novelty or Lean certification is asserted for those deductions.