bibkey: lengyel1995fibonacciorder authors: T. Lengyel year: 1995 title: The Order of the Fibonacci and Lucas Numbers doi: 10.1080/00150517.1995.12429139 url: https://fq.math.ca/Scanned/33-3/lengyel.pdf claim: The p-adic order formulas for F_n and L_n include v_p(F_(nk))=v_p(F_n)+v_p(k) for p outside {2,5}, positive n and k, and rank(p) dividing n, with separate formulas for two and five. strata_touched: [] license: citation-only triage: anchor

Fibonacci and Lucas p-adic orders

Verified locator

T. Lengyel, The Order of the Fibonacci and Lucas Numbers, The Fibonacci Quarterly 33(3) (1995), 234–239, https://doi.org/10.1080/00150517.1995.12429139 . Crossref identifies the author, title, volume, issue and pages. The journal scan https://fq.math.ca/Scanned/33-3/lengyel.pdf gives the two- and five-adic formulas in Section 2, Lemmas 1–2 (page 235), and the other prime valuations in the unnumbered theorem of Section 3 (page 236).

Claim and scope

Let alpha(p) be the least positive Fibonacci zero index and put e(p)=v_p(F_alpha(p)). For p outside {2,5}, the theorem gives v_p(F_n)=v_p(n)+e(p) if alpha(p) divides n, and zero otherwise. Consequently, for positive n,k with alpha(p) dividing n, v_p(F_(nk))=v_p(F_n)+v_p(k). The theorem also determines v_p(L_n). The initial valuation e(p) is retained, not replaced by one.

Lemma 1 gives v_5(F_n)=v_5(n) and v_5(L_n)=0 for n>0. Lemma 2 gives the separate two-adic cases: v_2(F_n)=0 when three does not divide n, one when n is three modulo six, and v_2(n)+2 when six divides n; v_2(L_n) is respectively zero, two and one in those three cases.

These formulas supply the classical square-divisibility threshold attributed inline in FPD.1 of Problems/wall-sun-sun-golden-unit-lift.md. The dossier provides a recurrence proof of that corollary. It does not claim the valuation theory as new or infer the existence of a WSS prime.

Rank support used in a Robin tail estimate

Page 236, Section 3 opens with the classical statement that for a prime p outside {2,5}, either F_(p-1) or F_(p+1) is divisible by p. Theorem A immediately below says that p | F_n exactly when alpha(p) | n. Together these give alpha(p) | p-1 or alpha(p) | p+1. This implication does not require the first valuation e(p) to be one.

FIB theory §§163–164 use these facts to group the prime factors of a single F_j by their common first zero index, and bound the Euler product of high-index groups. That Robin synthesis is not a theorem claimed in Lengyel’s paper. The existing Lean module D5.S3.Arith.FibonacciRank provides the corresponding entry-point and prime-bound interfaces.