bibkey: albagonzalez2012recurrencedivisibility authors: Juan Jose Alba Gonzalez, Florian Luca, Carl Pomerance, and Igor E. Shparlinski year: 2012 title: On numbers n dividing the nth term of a linear recurrence doi: 10.1017/S0013091510001355 url: https://doi.org/10.1017/S0013091510001355 claim: The proof of Theorem 1.2 bounds weighted prime appearance-index sums on dyadic intervals with decay of order the inverse square root of the interval scale. strata_touched: [] license: citation-only triage: anchor
Weighted appearance-index tails
The published article is in Proceedings of the Edinburgh Mathematical Society 55 (2012), pages 271–289. Its proof of Theorem 1.2 splits primes in dyadic intervals according to the size of their Lucas appearance index. The resulting weighted shell estimate has order inverse square root of the interval scale. The implicit constant depends on the recurrence; this argument does not state the Fibonacci constant 16 for every real cutoff at least two.
The FIB theory volume gives the explicit Fibonacci bound
where the sum is over primes and z(p) is the least positive index with
p dividing the Fibonacci number. The decay order is known from the published
shell argument. The explicit constants and the truncated Euler-logarithm
estimate in the volume are project deductions, without a novelty claim.
For integral cutoffs, bounding the product of distinct primes in one rank bucket bounds the number of primes above the cutoff. Combining this count with the complete bucket’s Euler-logarithm estimate yields the common square-root bound, then summing over the divisors of a Fibonacci index yields the truncated Euler tail. This last estimate is not attributed to the article.
Verified locator
- DOI and original text: https://doi.org/10.1017/S0013091510001355
- Scope: proof of Theorem 1.2, §4, printed pages 281–282.