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

A neutral schematic of the objects and relations in the statement.

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.

1Context

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

2Remarks

Remark 1. Arithmetic is modulo 65537.

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

3What 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

Current 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.

1Packet records

Record	Kind	Assessment
	Result	Reproduced
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 ReproducedA computation someone reran from the artifact on this page. 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 →
	Result	Established
claim · Theorem 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 EstablishedA complete argument is recorded and has been reviewed. 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 →
	Trace	Supported
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 →
	Artifact	Reproduced
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 ReproducedA computation someone reran from the artifact on this page. 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 →
	Artifact	Reproduced
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 ReproducedA computation someone reran from the artifact on this page. 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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-25. 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. The checked sources do not settle the full acceptance condition.

The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: 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.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.

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 connectTyped relations and evidence flow

ProblemLargest cycle of a Hénon permutation over the 65537 field

Computation 1A certified cycle has length 294,672in this packetReproduced
Artifact 1Exact cycle replay with an orbit digestverifiesReproduced
Artifact 2Complete p=1009 cycle decompositiontestsReproduced
Route 1The finite-field literature gives statistical contextinformsSupported
Theorem 1The map is a reversible area-preserving Hénon mapinformsEstablished

2See also

Prime exceptions to connectivity of the Markoff grapharithmetic dynamics
Rank log-concavity for symmetric binary matrices through order fiftyfinite fields
Trace-indistinguishable triples in sl2(F5)finite fields

How to cite

TheoremDB contributors, “Largest cycle of a Hénon permutation over the 65537 field,” TheoremDB research memory, snapshot of July 25, 2026. https://theoremdb.org/statements/henon-65537-max-cycle

This page as plain text: henon-65537-max-cycle.md

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

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. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.The map is a reversible area-preserving Hénon map. Coordinate swap reverses H, placing its finite-field cycles in the Roberts and Vivaldi setting. 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.Also cited at Nonlinearity 18 (2005), 2171-2192.Also cited at Direct identities above; Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192.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.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. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.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.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

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance