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This research defines a dynamical system using the iterated Fermat quotient map, aiming to understand its periodic orbits and their arithmetic significance.

April 8, 2026Open Access

Iterated Fermat Quotient Dynamics on (Z/pZ)×: Bridge Identities, Sub-Tower Truncation, and Obstructions

Key Points

This research defines a dynamical system using the iterated Fermat quotient map, aiming to understand its periodic orbits and their arithmetic significance.
Defined and studied the iterated generalized Fermat quotient map on (Z/pZ)×.
Proved six unconditional results regarding bridge identities and periodicity.
Analyzed structural gaps and determinacy in base-p digit representations.
Conducted tests to verify algebraic results up to p_max = 100,000.
Established general bridge identities relating tower digits recursively.
Showed that periodicity exists in towers modulo p^N and identified negative results about Cauchy sequences.
Documented the vanishing behavior of base-p digits linked to order within the sub-tower truncation.
Identified a fundamental obstruction related to quadratic-residue information.

Abstract

We define and study a dynamical system on (Z/pZ) × given by the iterated generalised Fermat quotient map Φ: b ↦ (b^ordₚ (b) −1) /p mod p. This map replaces the decimal reptend of 1/p in base b by a new base, generating eventually periodic orbits whose structure encodes arithmetic information about p. We prove six unconditional results: (A) a general bridge identity expressing the k-th tower digit aₖ as a recursive function of the sub-tower digits a₁^ (d), …, aₖ^ (d) of bᵈ via the relation aₖ ≡ n·aₖ^ (d) + Tₖ (a₁^ (d), …, a₊-₁^ (d) ; n) (mod p), where n = (p−1) /d and Tₖ collects binomial cross-terms, valid at all tower levels k ≥ 1; (B) a structural gap showing that every fixed point b* satisfies b*^ord (b*) ≡ 1+pb* (mod pN) for all N, with p-adic digit d₁ = 0; (C) tetration periodicity, establishing that the tower b*^kⁿ is eventually periodic modulo pN with period ord₏^₍−₁ (k), and documenting a negative result: the sequence is not Cauchy in Zₚ (νₚ = 1 constant) ; (D) a sub-tower truncation: for every element of order d, the base-p digits of bᵈ vanish at depth ≥ d, and a corollary showing that the tower digits aₖ for k ≥ d are deterministic functions of a₁, …, a₃−₁; (E) the sub-tower marginal zero count E#aₖ^{ (d) =0} → k+1 as p → ∞ (structural summand k unconditional via Dirichlet; equidistributional summand +1 heuristic for k ≥ 2) ; (F) a parity constraint forcing χ (b) = −1 ⇒ 2 | ordₚ (b). We decompose h (−p) = − (1/p) ΣO SO by orbits of Φ, prove the parity reduction Σ χ (a) qₚ (a) ≡ −Σ χ (a) a^−1 (mod p) (correcting a false vanishing claim in v0. 65), and document a negative result: the co-length sequence of an orbit does not determine its character-weighted contribution SO (verified at p=251: two 2-cycles sharing co-lengths (1, 1) yield S = −118 ≠ −272). This eliminates the route from co-lengths alone to h (−p) and identifies the positional (quadratic-residue) information as the irreducible obstruction. A companion computational report (DOI: 10. 5281/zenodo. 19422983) verifies all algebraic results to pₘax = 100, 000 (17, 228, 275 bridge tests at five tower levels, zero mismatches) and documents the statistical universality of Φ as a random function.

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