bibkey: marcus2025a003558 authors: N. J. A. Sloane; Michel Marcus year: 2025 title: “OEIS A003558, least m with 2^m = ±1 mod 2n+1, with Marcus’s tent-map cycle-length conjecture” doi: null url: https://oeis.org/A003558 claim: “%N Least number m > 0 such that 2^m == +-1 (mod 2n + 1). %C It appears that a(n) is the length of the cycle starting at 2/(2n+1) for the map x->1 - abs(2x-1). - Michel Marcus, Jul 16 2025” strata_touched:

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

OEIS A003558 and Marcus’s tent-map comment

The associated theorem identifies the least positive signed modular return of 2 with the least return time of numerator 2 under tent N k = if 2*k <= N then 2*k else 2*N - 2*k, for N=2n+1 and n>0. For N>0 and k<=N, division by N identifies these two branches with x -> 1 - abs(2*x-1) at x=k/N; natural subtraction does not truncate the second branch in this range. The even orbit from numerator 2 remains in that range.

Only the quoted Marcus sentence is settled. The entry’s comments about Kaprekar cycles, iteration of x^2-2, and the Kappraff-Adamson base conjecture are not claimed. The offset is %O 0,3, but n=0 is excluded: then N=1 and the starting point 2/N is outside [0,1].

Periodic orbits of the tent map on rationals are classical dynamical-systems material, while this cycle-length identity was not found stated in the searched literature described in the associated problem dossier.

Verified locator

The supplied source reading is dated 2026-09-15; independent online verification is ASSUMED-UNVERIFIED under the no-network constraint.

• URL: https://oeis.org/A003558
• %N (verbatim): Least number m > 0 such that 2^m == +-1 (mod 2n + 1).
• %C (verbatim): It appears that a(n) is the length of the cycle starting at 2/(2n+1) for the map x->1 - abs(2x-1). - Michel Marcus, Jul 16 2025
• %A (verbatim): N. J. A. Sloane
• %O (verbatim): 0,3