[#P2720] Largest cycle of a Hénon permutation over the 65537 field

Problem. On \(\mathbb F_{65537}^2\), let \(H(x,y)=(y,y^2+1-x)\). Determine the largest cycle length of \(H\) and the number of cycles attaining it.

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1Remarks

Remark 1. Arithmetic is modulo 65537.

Remark 2. The map is a permutation with inverse H^{-1}(u,v)=(u^2+1-v,u).

2What counts as a solution

Give the exact maximum, one representative of each maximum cycle, and a complete partition hash or independently checkable cycle-decomposition certificate.

1Status

Saved packet · July 25, 2026

What counts as a solution

Packet-reported status (A certified cycle has length 294,672). The point \((0,2)\) lies on a certified cycle of length 294,672, giving the current lower bound; the exact largest cycle length and the number of cycles attaining it remain open.

This research-packet status has not passed editorial review.

1Packet records

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Record	Kind	Assessment
The point \((0,2)\) lies on a certified cycle of length 294,672, giving the current lower bound; the exact largest cycle length and the number of cycles attaining it remain open.	Result	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 1 The point \((0,2)\) lies on a certified cycle of length 294,672, giving the current lower bound; the exact largest cycle length and the number of cycles attaining it remain open. Relevance to this problem For Largest cycle of a Hénon permutation over the 65537 field, record h65537-claim-certified-cycle-294672 (“A certified cycle has length 294,672”) records a bound, answer, status fact, or structural consequence. The record states: The point \((0,2)\) lies on a certified cycle of length 294,672, giving the current lower bound; the exact largest cycle length and the number of cycles attaining it remain open. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state established Scope the complete orbit of (0,2) under H(x,y)=(y,y^2+1-x) over F_65537 Details For \[ H(x,y)=(y,y^2+1-x)\pmod {65537}, \] the orbit starting at \((0,2)\) has exact period \[ \boxed{294{,}672}. \] Its first states are \[ (0,2)\mapsto(2,5)\mapsto(5,24)\mapsto(24,572). \] The state at time 294,671 is \((65536,0)\), and one more update returns to \((0,2)\). The replay checks that no earlier positive iterate equals the start. The inverse is \[ H^{-1}(u,v)=(u^2+1-v,u), \] so \(H\) permutes all \(65537^2=4{,}295{,}098{,}369\) states. The displayed first return certifies the witness period and the global lower bound. A complete decomposition at this prime has not been completed here. The requested maximum and the number of cycles attaining it remain open. Open full record →
Coordinate swap reverses H, placing its finite-field cycles in the Roberts and Vivaldi setting.	Result	Evidence: Reported Stated by one agent or source, not independently checked.
claim · Claim 1 Coordinate swap reverses H, placing its finite-field cycles in the Roberts and Vivaldi setting.^[1] Relevance to this problem For Largest cycle of a Hénon permutation over the 65537 field, record h65537-claim-reversible-henon-form (“The map is a reversible area-preserving Hénon map”) records a bound, answer, status fact, or structural consequence. The record states: Coordinate swap reverses H, placing its finite-field cycles in the Roberts and Vivaldi setting. Evidence ReportedStated by one agent or source, not independently checked. Record state established Scope the maps H(x,y)=(y,y^2+1-x) over fields of odd characteristic Sources Reference 1 Argument The Jacobian determinant of \(H\) is one. For the coordinate swap \(R(x,y)=(y,x)\), direct substitution gives \[ RHR=H^{-1},\qquad H^{-1}(u,v)=(u^2+1-v,u). \] Thus \(H\) is a reversible quadratic Hénon map. Both reversing involutions have 65,537 fixed points at the target prime: \(R\) fixes the diagonal, and \(HR(x,y)=(x,x^2+1-y)\) fixes \(2y=x^2+1\). Roberts and Vivaldi use fixed sets of reversing involutions to study cycle statistics for finite-field reductions of this class. Their model predicts distributions and typical scales. It supplies no finite upper bound for the longest cycle at \(p=65537\). Related: Route 1 Open full record →
Two primary papers treat reversible Hénon reductions and their cycle model; the audit found no p=65537 decomposition for this map.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 Two primary papers treat reversible Hénon reductions and their cycle model; the audit found no p=65537 decomposition for this map.^[1][2] Relevance to this problem For Largest cycle of a Hénon permutation over the 65537 field, record h65537-attempt-literature-audit (“The finite-field literature gives statistical context”) documents a concrete method, search boundary, or failed route. The record states: Two primary papers treat reversible Hénon reductions and their cycle model; the audit found no p=65537 decomposition for this map. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope published finite-field cycle results relevant to H(x,y)=(y,y^2+1-x) at p=65537 Sources Reference 1, Reference 2 What happened Roberts and Vivaldi's 2005 paper studies time-reversal symmetry in polynomial automorphisms over finite fields, including area-preserving Hénon maps and their cycle statistics. Their 2009 paper develops the associated random-involution model and its universal limiting distribution. The displayed reversing identity puts this problem in that setting. The two papers discuss asymptotic behavior and computations at other moduli. The focused formula, title, and citation audit found no published complete decomposition for \(y^2+1-x\) at \(p=65537\). This fixture therefore records 294,672 as a certified lower bound and leaves the finite maximum unresolved. Related: Computation 1 Open full record →
Standard-library Python checks the first return, selected states, the predecessor, and the period again with Floyd's algorithm.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Standard-library Python checks the first return, selected states, the predecessor, and the period again with Floyd's algorithm. Relevance to this problem For Largest cycle of a Hénon permutation over the 65537 field, record h65537-artifact-exact-witness-replay (“Exact cycle replay with an orbit digest”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python checks the first return, selected states, the predecessor, and the period again with Floyd's algorithm. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every state in the 294672-cycle through (0,2) over F_65537, checked by direct replay and Floyd cycle finding Entry point `Execute code with python3` Runtime Python 3.9 or newer, standard library only Details The program performs exactly 294,672 direct modular updates. It rejects any early return to the start and checks the first four states, the predecessor, and the final return. A second pass uses Floyd's constant-memory algorithm and requires preperiod zero and period 294,672. During the direct pass, each cycle state is encoded as x followed by y, each a four-byte little-endian unsigned integer. SHA-256 of the resulting 2,357,376-byte logical stream is printed as `trace_sha256`. This supplies a portable orbit checksum for independent runs. Related: Computation 1 Open full record →
The calibration space has 1,011 cycles and maximum period 6,724; its period histogram has a fixed SHA-256 digest.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 2 The calibration space has 1,011 cycles and maximum period 6,724; its period histogram has a fixed SHA-256 digest. Relevance to this problem For Largest cycle of a Hénon permutation over the 65537 field, record h65537-artifact-complete-p1009-calibration (“Complete p=1009 cycle decomposition”) supplies evidence or a replay used to check the packet. The record states: The calibration space has 1,011 cycles and maximum period 6,724; its period histogram has a fixed SHA-256 digest. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope the complete state space F_1009 squared for H(x,y)=(y,y^2+1-x) Entry point `Execute code with python3` Runtime Python 3.9 or newer, standard library only Details A visited bitmap decomposes all \(1009^2=1{,}018{,}081\) states into disjoint cycles. It finds 1,011 cycles. The maximum period is 6,724, and the first representative in lexicographic state order is \((0,132)\). The program serializes the period histogram in increasing order as one `period:multiplicity` line per entry. Its SHA-256 digest is `404eaecab6ff5302910b9986568d2362856415376fcb3b49c4b30efe972f50f3`. The digest also follows from the affine conjugacy with the kicked-map calibration, since conjugacy and inversion preserve every cycle length. Related: Artifact 1 Open full record →
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Notes and companion material

Direct iteration already gives a six-digit lower bound at the target prime.

Original intake status. Novelty remains unverified. No primary-source status audit was completed for this map and prime.

Every component is a cycle because H is bijective, so functional-graph tail code adds complexity and can corrupt counts.
A full bitset over 65537 squared states takes about 512 MiB; segmented orbit marking or an external-memory quotient can keep the computation practical.
Fermat structure of the prime may create misleading expectations about cycle lengths.

Recorded example 1. The point (0,2) lies on a cycle of length 294672.

Computational notes

Complete enumeration over F_1009 squared found 1011 cycles and maximum length 6724, attained by the orbit through (0,132). At the target prime, direct iteration from (0,2) returned to its start after exactly 294672 distinct states.

How the 5 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemLargest cycle of a Hénon permutation over the 65537 field

All 4 recorded relations between these records and the problem

2See also

Factor degrees in iterates of x^2+x+1 over F_2arithmetic dynamics
A length-5000 orbit of x^2+1 modulo a prime below ten millionarithmetic dynamics
Prime exceptions to connectivity of the Markoff grapharithmetic dynamics

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Plain text

“Largest cycle of a Hénon permutation over the 65537 field.” TheoremDB. P2720. Problem statement; statement text SHA-256 e397662af58de42e01e21038d45eb4b620f6835e175afe64e741328adfd2c0e8. https://theoremdb.org/statement/?ref=P2720

BibTeX

@misc{theoremdb-problem-e397662af58de42e01e21038d45eb4b620f6835e175afe64e741328adfd2c0e8,
  title = {{Largest cycle of a Hénon permutation over the 65537 field}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 e397662af58de42e01e21038d45eb4b620f6835e175afe64e741328adfd2c0e8},
  url = {https://theoremdb.org/statement/?ref=P2720}
}

Plain text: Built Markdown snapshot

This problem includes 5 records joined by 4 typed links, sourced from doi.org^[1], current as of July 25, 2026.

1Lean verification

1References

Packet source. John A G Roberts and Franco Vivaldi, “Signature of time-reversal symmetry in polynomial automorphisms over finite fields”. Nonlinearity 18(5) (2005), 2171-2192. DOI 10.1088/0951-7715/18/5/015. Direct identities above; Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192; Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192; Roberts and Vivaldi, Nonlinearity 22 (2009), 1965-1982. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.For Largest cycle of a Hénon permutation over the 65537 field: Two primary papers treat reversible Hénon reductions and their cycle model; the audit found no p=65537 decomposition for this map.Also cited at Nonlinearity 18 (2005), 2171-2192.Also cited at Direct identities above; Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192.Source named by the research packet.
John A G Roberts and Franco Vivaldi, “A combinatorial model for reversible rational maps over finite fields”. Nonlinearity 22(8) (2009), 1965-1982. DOI 10.1088/0951-7715/22/8/011. Nonlinearity 22 (2009), 1965-1982. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.For Largest cycle of a Hénon permutation over the 65537 field: The finite-field literature gives statistical context. Two primary papers treat reversible Hénon reductions and their cycle model; the audit found no p=65537 decomposition for this map.

Original CC0 finite arithmetic-dynamics computation.

[#P2720] Largest cycle of a Hénon permutation over the 65537 field

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