The eventual period of Γ(k,Fₙ): ninety new values and a closed form in Pisano periods

Abstract

For relatively prime positive integers a,b exactly one of the two Diophantine equations ax+by=(a-1)(b-1)/2 and 1+ax+by=(a-1)(b-1)/2 has a nonnegative integral solution, and Γ(a,b) ∈ {0,1} records which one. Chu, Miller and Tresch (arXiv:2512.12681) write Tₖ((aₙ)) for the eventual period of (Γ(k,aₙ))_(n ≥ 1), prove that Tₖ((Fₙ)) divides the Pisano period π(2k), print Tₖ((Fₙ)) for k ≤ 10, and ask for its value for every k. We compute the ninety values 11 ≤ k ≤ 100 and verify, for every k ≤ 200 by machine-checked computation from the Diophantine definition and for every k ≤ 2000 by an independent computation, that Tₖ((Fₙ))=π(Pₖ), where Pₖ (=k for odd k, 2k for even k) is the period of Γ(k, ·) on ℕ; that is, Tₖ((Fₙ))=π(k) for odd k and π(2k) for even k. This value is strictly smaller than π(2k) for 60 of the first 200 values of k, among them k=3 of the printed table, and the set of k with Tₖ((Fₙ))=2π(k) is not the set of powers of two (k=40 is the least counterexample) but is described by a 2-adic condition on the Pisano period of the odd part of k. For the companion problem on arithmetic progressions pn-r we note that the coprime case gcd(k,p)=1 follows for every k from the source's own periodicity theorem, and exhibit T₁₅((5n-1))=1 and T₆((2n))=3 ≠ 6=T₆((2n-1)), which rule out the natural candidate formulas in the non-coprime range. The closed form for all k is stated as a conjecture. Every computational statement is verified in Lean 4 by evaluated computation, without Mathlib.

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