slug: pisano-order-range-refutation bibkey: benfieldlippard2025pisanozeros doi: null url: https://arxiv.org/abs/2407.20048v2 triage: theorem motivation_gids:

• D5/S3/Arith/PisanoOrderRangeRefutation.result

The Order of a Pisano Period Is Not Confined to Two

Problem

Benfield and Lippard, Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers, arXiv:2407.20048v2, 30 January 2025, close with Conjecture 5.3 on the range of the order as the parameters of a two-term recurrence vary. Clause (v) reads, from the LaTeX source fiborder.tex:

\item $\{0,1,2\}$ if $b \neq \pm1$ and $|a|-|b| = 1$,

Section 4 defines the objects: F_0 = 0, F_1 = 1, F_n = a F_{n-1} + b F_{n-2}. Section 1 defines the order: “The number of zeros in a Pisano period is the order of m, denoted omega(m).” Section 5 fixes the degenerate convention: “define the order to be zero if the sequence is eventually periodic modulo m and this period contains no multiples of m.”

Written out, the assertion is that for all integers a, b and all m > 1, if b ≠ ±1 and |a| - |b| = 1, then omega_(a,b)(m) ∈ {0,1,2}.

Motivation

The frozen theorem D5/S3/Arith/PisanoOrderRangeRefutation.result refutes it.

Gap

Issue 9448 records the screen carried out before the probe. The conjecture does not appear in the repository’s screening records, under Problems/, D5/ or Library/; no follow-up on arXiv cites it. Citation indices were not exhaustively reachable, so this is a bounded negative finding.

Route

The repository already carries the machinery, so the settlement reuses it rather than rebuilding it. Its Lucas convention is x_{n+1} = p x_n - q x_{n-1}, so the paper’s (a,b) is (p,q) = (a,-b). D5/S1/Recurrence/LucasEvenDescent supplies the sequence, the entry point and the theorem that the zero indices are exactly the multiples of the entry point; D5/S1/Recurrence/LucasCompanion supplies the period as the order of the companion matrix. Because the zeros are exactly those multiples and a period ends on a zero, the number of zeros in one period is the period divided by the entry point.

Take (a,b) = (3,2) and m = 13, so p = 3 and q = -2, a unit modulo 13 with inverse 6. The terms are 0, 1, 3, 11, 0, 9, 1, 8, 0, 3, 9, 7, and the pair of consecutive terms then returns to (0,1), so the period is 12. The entry point is 4: the fourth term is 3 · 11 + 2 · 3 = 39, a multiple of 13, while the first three are 1, 3 and 11. The order is 12 / 4 = 3, outside {0,1,2}, while b = 2 is not ±1 and |3| - |2| = 1.

Falsifier

A different value for any of the first thirteen terms modulo 13, a different entry point, or a period other than 12 would invalidate the witness. The term at index 12 is 3 · 7 + 2 · 9 = 39 and the term at index 13 is 3 · 0 + 2 · 7 = 14 ≡ 1, which is what closes the period at 12.

Evidence

The failure is not isolated and its cause is structural. Finiteness of the order in this region comes from x^2 - a x - b having ±1 among its roots: b = a+1 gives roots a+1 and -1, b = 1-a gives roots 1 and -a, and in either case the sequence has a closed form whose vanishing is governed by a single multiplicative order, so a period carries at most two zeros. In absolute values that family is |a| - |b| = 1 when b < 0 and |b| - |a| = 1 when b > 0. Clause (v) keeps only the first shape and so admits the pairs (3,2), (4,3), (5,4), …, whose characteristic polynomials have irrational roots — the discriminant at (3,2) is 17.

Measured, over 2 ≤ m ≤ 2000 unless stated otherwise:

The first row is the example the paper itself reports having checked; it satisfies |b| - |a| = 1 and does not satisfy the condition as printed. The computation was checked against published facts before being trusted: for the ordinary Fibonacci sequence it returns exactly {1,2,4} over all m ≤ 4000, the classical theorem the paper’s abstract quotes; it returns a period of 20 with four zeros at m = 5, matching the worked example in Section 4; and a period of 60 at m = 10, Lagrange’s 1877 reading quoted in Section 1.

Smaller witnesses with the same shape: (a,b) = (7,6) at m = 5, period 12, and (4,3) at m = 5, period 24.

Triage

theorem; Tier 1 named external conjecture in the closing section of a preprint, preregistered in issue 9448 before the probe. The admission basis is open-problem-resolution; the conservative classification is proof_shape: bind-only with escape_witness: none, since the settlement rests on instantiating frozen prerequisites at a concrete witness. The computational use is a certified-instance with a typed refutes edge from result to claim.

ASSUMED-UNVERIFIED

The formal claim is the printed clause restricted to moduli where b is invertible, that is, to the case where the sequence is periodic from the start. The restriction weakens the claim, so refuting it refutes the clause as published; no separate treatment of the degenerate moduli is offered here.

This settlement refutes clause (v) as published. It does not prove the corrected clause: the statement that the order stays in {0,1,2} whenever ±1 is a root of the characteristic polynomial is supported here only by the measured ranges above, not by a proof.

The literature screen is bounded: the preprint was read in full and no follow-up citing Conjecture 5.3 was reachable; citation-index result pages were not exhaustively reachable, so no worldwide priority claim is made.