bibkey: lang2018a319197 authors: Wolfdieter Lang year: 2018 title: “OEIS A319197, factors of Fibonacci(2^(n-2)3m), with the dyadic divisibility conjecture” doi: null url: https://oeis.org/A319197 claim: “%N All entries from a(3) to a(n) appear in addition to 2^n as factors in the conjectured factorization of Fibonacci(2^(n-2)3m) for n >= 3 and all m >= 0. %C It appears that Fibonacci(2^(n-2)3m)/(2^n) is a nonnegative integer for n >= 3 and all m >= 0.” strata_touched:

• D5/S1/Recurrence/LangFibDyadicIndexDivisibility license: citation-only triage: anchor

OEIS A319197

The associated theorem proves only the first COMMENT sentence: for all natural numbers n and m, if 3 <= n, then 2^n divides Nat.fib (2^(n-2)*3*m). Natural-number subtraction in n-2 agrees with ordinary subtraction under this hypothesis. Because 2^n is positive, this divisibility gives the stated nonnegative-integral quotient.

The NAME supplies context, not an additional proved claim. The product A(n) = Product_{j=3..n} a(j), the I(n; m) factorization conjecture, and the specific factors from A049660 and A253368 are not claimed.

Lengyel (1995), “The order of the Fibonacci and Lucas numbers”, Fibonacci Quarterly 33, gives the general exact 2-adic valuation v_2(F_k) as prior literature; this module proves the divisibility directly by induction and does not claim the exact valuation or priority over that result.

Verified locator

The following OEIS lines were checked against https://oeis.org/A319197 on 2026-09-15.

• URL: https://oeis.org/A319197
• %N (verbatim): All entries from a(3) to a(n) appear in addition to 2^n as factors in the conjectured factorization of Fibonacci(2^(n-2)3m) for n >= 3 and all m >= 0.
• %C first sentence (verbatim): It appears that Fibonacci(2^(n-2)3m)/(2^n) is a nonnegative integer for n >= 3 and all m >= 0.
• %A (verbatim): Wolfdieter Lang, Oct 09 2018

Separate sharpness clause and the corrected normal form

The FORMULA field additionally says that there is no further uniform factor greater than one, citing I(n, 1) = 1. This is a different claim from the first COMMENT sentence above. At n = 7, its supplied values give

$$D_7=2^7\cdot 1\cdot 9\cdot 161\cdot 51841\cdot 6989569=67205083036226688,$$

whereas exact arithmetic gives

$$F_{96}=51680708854858323072=769D_7.$$

Proposition 6 of Jeffery–Pereira gives F_96 | F_(96m) for every positive integer m, and F_0 = 0 covers m = 0. Consequently 769 D_7 divides every F_(96m). This refutes the separately stated sharpness clause at n = 7; it does not invalidate the proved dyadic divisibility clause.

The corrected layers

$${\ell }_0=9,{\ell }_{j+1}=2{\ell }_j^2-1$$

are already recorded as A081459(j+2). Artur Jasinski entered the recurrence in 2008, and Ehren Metcalfe entered A081459(n)=L(3*2^(n-1))/2 in 2017. In particular,

$$6989569\cdot 769=5374978561={\ell }_3.$$

OEIS A081460 records A081460(k+2)=F_(6*2^k)/2 and the corresponding Lucas product. Together with A081459, this gives

$$F_{6\cdot 2^k}=2^{k+3}\prod _{0\le j<k}{\ell }_j.$$

The recurrence, Lucas interpretation, Newton iteration, and product formula are cited results rather than new claims of this repository.

The ordinary character/rank follow-up uses standard Fibonacci rank and valuation inputs; it does not give a Wall-Sun-Sun example. The elementary correction concerns the supplied A319197 data and its explicit best-possible assertion. It does not define the unprovided infinite tail of that entry or refute integrality at every later level.

Verified source locators: A319197 version 7, dated 2018-10-25; A081459 version 44, dated 2025-06-23; and A081460 version 46, dated 2026-04-09. The uniform-divisor characterization follows from Jeffery–Pereira, ISRN Algebra (2014), Article 750325, DOI 10.1155/2014/750325, Proposition 6; their Theorem 7 gives the corresponding Fibonacci gcd law. The entry’s power-of-two comment is settled by Lengyel, Fibonacci Quarterly 33(3) (1995), 234–239, Lemma 2, as restated in Medina–Rowland, Fibonacci Quarterly 53(3) (2015), 265–271, Theorem 1.4.