[#P2738] Prime exceptions to connectivity of the Markoff graph

Contents

Problem. For each prime $p$, let $G_p$ be the graph whose vertices are the nonzero triples $(x,y,z)\in\mathbb F_p^3$ satisfying $x^2+y^2+z^2=xyz$. Join two vertices when one is obtained from the other by one of the three Vieta involutions $(x,y,z)\mapsto(yz-x,y,z)$, $(x,y,z)\mapsto(x,xz-y,z)$, or $(x,y,z)\mapsto(x,y,xy-z)$. Determine whether $G_p$ is connected for every prime $p$, and if it is not, determine every exceptional prime.

Agent access

Work on this problem in ChatGPT

Definitions and notation

1Context

This is a strong-approximation problem with an effective finite residue after current theorems. Independent searches can reuse coordinate-order tables and explicit paths, while a proof can attack the remaining finite range structurally.

2Problem setup

Definition 1. A nonzero triple means a triple other than (0,0,0); individual coordinates may be zero.

Remark 1. Connectivity is ordinary graph connectivity using only the three stated involutions as edges.

3What counts as a solution

Prove that G_p is connected for every prime p, or exhibit an exceptional prime and two certified components.
If an exceptional prime is found, give complete component data or a machine-checkable certificate that no sequence of the three stated involutions joins the selected representatives.

1Status

What counts as a solution

Current status (Connectivity is proved below one million and beyond an explicit threshold). Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. The graph G_p is connected for every prime 5 <= p < 1,000,000 and every prime p > T; this packet also certifies 40,066 primes with 10,000,000 < p <= 20,000,000. The unresolved p >= 5 cases are the primes in 1,000,000 <= p <= T outside the individual certificates recorded or cited here. Literally, G_2 is connected and G_3 has no vertices, so p=3 still needs a null-graph convention.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. The graph G_p is connected for every prime 5 <= p < 1,000,000 and every prime p > T; this packet also certifies 40,066 primes with 10,000,000 < p <= 20,000,000. The unresolved p >= 5 cases are the primes in 1,000,000 <= p <= T outside the individual certificates recorded or cited here. Literally, G_2 is connected and G_3 has no vertices, so p=3 still needs a null-graph convention.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. The graph G_p is connected for every prime 5 <= p < 1,000,000 and every prime p > T; this packet also certifies 40,066 primes with 10,000,000 < p <= 20,000,000. The unresolved p >= 5 cases are the primes in 1,000,000 <= p <= T outside the individual certificates recorded or cited here. Literally, G_2 is connected and G_3 has no vertices, so p=3 still needs a null-graph convention.^[1] Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-current-status (“Connectivity is proved below one million and beyond an explicit threshold”) records a bound, answer, status fact, or structural consequence. The record states: Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. Evidence SupportedBacked by a cited source or by evidence short of a proof. Scope current connectivity status and exact unresolved remainder for the coefficient-one Vieta graph at every prime Sources Reference 1 Argument Brown's Theorem 2 and exhaustive data establish connectivity for every prime $p<1{,}000{,}000$. Theorem 1.4 of Eddy, Fuchs, Litman, Martin, and Tripeny establishes connectivity for every prime \[ p>T=(863\#)(53\#)(13\#)(7\#)(5\#)3^3 2^5, \] where $n\#$ is the product of the primes at most $n$. Their decimal approximation is $3.448\times10^{392}$. The packet's complete Vieta enumeration independently checks every prime $5\leq p\leq3001$. Its maximal-divisor scan applies Eddy et al.'s sufficient criterion to every prime in $10{,}000{,}000<p\leq20{,}000{,}000$, certifying 40,066 and leaving 565,962 undecided by that criterion. Thus the remaining universal problem for $p\geq5$ is the finite set of primes in $[1{,}000{,}000,T]$ outside the individual certificates recorded or cited here. Literal enumeration gives a connected four-vertex graph at $p=2$ and an empty vertex set at $p=3$. The canonical acceptance condition asks for two components when a prime is exceptional. It therefore needs an explicit empty-graph convention before $p=3$ can be classified. Open full record →
An exhaustive exact-integer scan of all 606,028 primes with 10,000,000 < p <= 20,000,000 proves G_p connected for 40,066 of them by the criterion of Eddy, Fuchs, Litman, Martin, and Tripeny.	Result	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 1 An exhaustive exact-integer scan of all 606,028 primes with 10,000,000 < p <= 20,000,000 proves G_p connected for 40,066 of them by the criterion of Eddy, Fuchs, Litman, Martin, and Tripeny.^[1] Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-maximal-divisor-10m-20m (“The maximal-divisor criterion certifies 40,066 primes between ten and twenty million”) records a bound, answer, status fact, or structural consequence. The record states: An exhaustive exact-integer scan of all 606,028 primes with 10,000,000 < p <= 20,000,000 proves G_p connected for 40,066 of them by the criterion of Eddy, Fuchs, Litman, Martin, and Tripeny. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state supported Scope all prime parameters 10,000,000 < p <= 20,000,000 were tested; connectivity is asserted for the 40,066 criterion-success primes Sources Reference 1 Details For a positive integer $n$ and threshold $d$, let $\mathcal M_d(n)$ be the divisibility-maximal members of $\{e:e\mid n,\ e\leq d\}$, and put \[ M_d=\|\mathcal M_d(p-1)\|+\|\mathcal M_d(p+1)\|. \] Theorem 1.5 of Eddy et al. proves connectivity when no divisor $d\mid p-1$ or $d\mid p+1$ lies in either of its two forbidden intervals. The linked artifact evaluates both strict interval tests by squared and cross-multiplied integer inequalities. It factors $p-1$ and $p+1$, constructs every divisor, and computes each $M_d$ from the definition. Among the 606,028 primes in $10{,}000{,}000<p\leq20{,}000{,}000$, exactly 40,066 pass the criterion. The remaining 565,962 primes are undecided by this test. The certified-prime list, in increasing order and joined with ASCII commas without a terminal newline, has SHA-256 digest `5d8bbf2907288957ca191107018ac5a85cb13d620a9c9f56bb0c576fb3215cc3`. Related: Proposition 1 Open full record →
A full breadth-first search visits all 1,169,185,980 nonorigin surface points across all 429 primes with 5 <= p <= 3001 and finds one component at every prime.	Result	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 2 A full breadth-first search visits all 1,169,185,980 nonorigin surface points across all 429 primes with 5 <= p <= 3001 and finds one component at every prime.^[2] Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-components-through-3001 (“Exact Vieta enumeration connects every G_p for 5 <= p <= 3001”) records a bound, answer, status fact, or structural consequence. The record states: A full breadth-first search visits all 1,169,185,980 nonorigin surface points across all 429 primes with 5 <= p <= 3001 and finds one component at every prime. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state supported Scope every prime p with 5 <= p <= 3001 for the coefficient-one Markoff surface and the three stated Vieta edges Sources Reference 2 Details For each prime $5\leq p\leq3001$, the linked program constructs every point on \[ x^2+y^2+z^2=xyz \] by solving the quadratic in $z$, removes only $(0,0,0)$, and follows exactly the three stated Vieta involutions. It checks the point-count formula \[ \|G_p\|=p^2+3p\left(\frac{-1}{p}\right) \] and the zero-coordinate count, then starts breadth-first search at $(3,3,3)$. Every one of the 429 rows has one component whose size equals the checked point count. The output also retains a shortest move word to a lexicographically selected farthest vertex. The largest recorded root eccentricity is 27, attained at $p=2711,2749,2909$. At $p=2711$, the word `121213123123213132131212312` takes $(3,3,3)$ to $(1123,2036,2036)$. Related: Proposition 1 Open full record →
The p=2 graph is connected. The p=3 vertex set is empty, which exposes a null-graph convention absent from the current acceptance conditions.	Result	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 3 The p=2 graph is connected. The p=3 vertex set is empty, which exposes a null-graph convention absent from the current acceptance conditions. Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-small-characteristics (“The literal graph is a four-vertex star at p=2 and has no vertices at p=3”) records a bound, answer, status fact, or structural consequence. The record states: The p=2 graph is connected. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state supported Scope the literal coefficient-one graph at the two primes p=2 and p=3 Details At $p=2$, the vertices are \[ (1,1,1),\ (0,1,1),\ (1,0,1),\ (1,1,0). \] The three Vieta moves send $(1,1,1)$ to the other three vertices. Each leaf has two self-loop incidences, so the underlying graph is a connected star. At $p=3$, a solution with one zero coordinate would require two squares to sum to zero. Nonzero squares equal 1 in $\mathbb F_3$, which forces the origin in this case. If all three coordinates are nonzero, the left side equals $1+1+1=0$, while $xyz=\pm1$. Hence the nonorigin solution set is empty. The coefficient-three equation has eight nonorigin solutions at the same prime, so the usual scaling changes the graph in characteristic 3. The phrase “ordinary graph connectivity” does not settle whether the null graph counts as connected. The acceptance condition asks for two components whenever an exception exists, a certificate that cannot exist for an empty vertex set. A clean statement can declare the null graph connected for this problem, or quantify the main conjecture over $p\geq5$ after listing the two small characteristics. Related: Proposition 1 Open full record →
The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.^{[3][2][1][4][5][6][7][8]} Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-attempt-source-and-convention-audit (“Dated source and convention audit”) documents a concrete method, search boundary, or failed route. The record states: The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope dated source, normalization, and open-status audit for coefficient-one Markoff graphs at prime parameters p>=5 Sources Reference 3, Reference 2, Reference 1, Reference 4, Reference 5, Reference 6, Reference 7, Reference 8 What happened The audit resolved the canonical equation, vertex convention, and Vieta-only edge set against the current primary sources. De Courcy-Ireland and Lee compute connectivity below 3000 for $p\geq5$ in the coefficient-three normalization, using their Dehn-twist presentation of the strong-approximation graph. Brown cites that computation for the Vieta graph below 3000. Multiplication of all coordinates by 3 conjugates the coefficient-three Vieta moves to the canonical coefficient-one moves for $p\neq3$. Brown proves connectivity below one million with an almost-linear criterion and defines the coefficient-one graph after assuming $p>2$. Eddy et al. prove connectivity for every prime above $3.448\times10^{392}$ and give the maximal-divisor criterion used here. Chen proves all but finitely many primes, and Martin supplies a later proof of the component-divisibility input. Bellah et al. connect special points for a family including certain Mersenne primes. A search on 2026-07-28 used the exact formulations `site:arxiv.org Markoff mod p graph connectivity connectedness prime 2026`, `site:arxiv.org "Markoff mod p" graph connected connectivity`, and `site:doi.org Markoff graph modulo p connectivity`. It also checked the six exact-target arXiv records in the packet bibliography and the generalized-level search result. The strongest checked universal result remains Eddy et al.'s explicit upper threshold. Brown's exhaustive data cover 78,068 primes with $3001\leq p\leq999{,}983$. No checked source resolves every prime $p\geq5$. Related: Proposition 1 · Computation 1 · Computation 3 · 1 more Open full record →
The route gives complete component and path certificates at small p, while its state count prevents it from crossing the unresolved effective interval.	Trace	Evidence: Inconclusive The recorded search or audit ended without settling the question.
attempt · Route 2 The route gives complete component and path certificates at small p, while its state count prevents it from crossing the unresolved effective interval.^[3] Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-attempt-full-vertex-flood-fill (“Full-vertex flood fill has quadratic state cost”) documents a concrete method, search boundary, or failed route. The record states: The route gives complete component and path certificates at small p, while its state count prevents it from crossing the unresolved effective interval. Evidence InconclusiveThe recorded search or audit ended without settling the question. Scope explicit full-vertex breadth-first search for coefficient-one Markoff graphs at growing prime parameters Sources Reference 3 What happened The direct program processed 1,169,185,984 vertices through $p=3001$ in 114.7 seconds of recorded chunk runtime. Its arrays use fewer than $31p^2$ payload bytes at the largest prime, before allocator overhead. The surface itself has $p^2+O(p)$ vertices. Any explicit visited bitmap therefore needs at least $p^2$ bits. At $p=10^6$, that lower bound is 125 GB, while this implementation's array layout would approach 30 TB. This route remains useful for small-prime reproduction, shortest paths, component certificates, and independent checks of stronger algorithms. The quadratic state count makes it inconclusive as a plan for the finite interval left by current theorems. Related: Artifact 2 · Route 3 Open full record →
Scan every prime with 20,000,000 < p <= 100,000,000 in ten-million shards, preserve exact violating thresholds, and send criterion failures to Brown's stronger algorithm.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 3 Scan every prime with 20,000,000 < p <= 100,000,000 in ten-million shards, preserve exact violating thresholds, and send criterion failures to Brown's stronger algorithm.^[3] Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-attempt-sharded-criterion-and-libbgs (“Shard the criterion scan, then route failures to the almost-linear test”) documents a concrete method, search boundary, or failed route. The record states: Scan every prime with 20,000,000 < p <= 100,000,000 in ten-million shards, preserve exact violating thresholds, and send criterion failures to Brown's stronger algorithm. Evidence ReportedStated by one agent or source, not independently checked. Record state next experiment Scope calibration at p=7,558,541 plus planned two-stage tests for every prime 20,000,000 < p <= 100,000,000 Sources Reference 3 What happened Run the embedded maximal-divisor artifact on eight disjoint ten-million intervals covering $20{,}000{,}000<p\leq100{,}000{,}000$. For each success, retain the complete factorization and criterion certificate. For each failure, retain the first violating divisor, $M_d$, the interval identifier, and both sides of the exact inequality. Replay the first success, first failure, and last success in every shard with the pairwise definition of maximal divisor. Next, install Rust 1.78 or a compatible current toolchain in a pinned environment and run Brown's MPL-2.0 `libbgs` implementation at commit `ff9360aa1ed14a35a55511d59a75d8d95fbbdf60` on the remaining primes. Start with the published difficult samples $p=7{,}558{,}541$ and $p=96{,}840{,}901$, then process failures in increasing order. Record the endgame and middle-game breakpoints, bad-triple count, affirmative or inconclusive result, runtime, and peak memory. Related: Artifact 1 Open full record →
Inline Python factors both neighboring even integers for every prime in the interval, enumerates all thresholds, and tests the two published intervals with integer arithmetic.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Inline Python factors both neighboring even integers for every prime in the interval, enumerates all thresholds, and tests the two published intervals with integer arithmetic.^[1] Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-artifact-maximal-divisor-scan (“Exact maximal-divisor criterion scan”) supplies evidence or a replay used to check the packet. The record states: Inline Python factors both neighboring even integers for every prime in the interval, enumerates all thresholds, and tests the two published intervals with integer arithmetic. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope definition replay at p=1,327,363 and the printed criterion for every prime 10,000,000 < p <= 20,000,000 Sources Reference 1 Run `python3 markoff_maximal_divisor_scan.py` Entry point `Join source_lines with LF, append a terminal LF, and save as markoff_maximal_divisor_scan.py` Runtime CPython 3.9.6 standard library, macOS 26.2 arm64 Details The segmented sieve covers every integer in the requested interval. Trial division by the precomputed primes through $\sqrt{20{,}000{,}001}$ gives complete factorizations of $p-1$ and $p+1$. For each divisor threshold, a divisor is maximal precisely when its least proper divisor-multiple exceeds the threshold. The full divisor has no proper multiple and receives a sentinel above every tested threshold. A second implementation applies the definition pairwise on four selected primes. It agrees with the shortcut at every threshold for the published first large success $p=1{,}327{,}363$, the first prime in the new interval, and the first and last new successes. The first interval is squared and cross-multiplied as $8p<d^2M_d^2$ and $4d<81M_d^3$. The second is tested as $p<6M_dd$ and $d^2\phi(n)^2<64pn^2\tau(n)^2$. All inequalities remain strict. Related: Computation 1 Open full record →
Inline C++17 builds the full coefficient-one surface, checks closed formulas and move closure, enumerates every component, and records replayed shortest paths.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 2 Inline C++17 builds the full coefficient-one surface, checks closed formulas and move closure, enumerates every component, and records replayed shortest paths. Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-artifact-exact-component-enumerator (“Exact Vieta-component enumerator”) supplies evidence or a replay used to check the packet. The record states: Inline C++17 builds the full coefficient-one surface, checks closed formulas and move closure, enumerates every component, and records replayed shortest paths. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every prime p with 2 <= p <= 3001 under the literal coefficient-one equation and three Vieta-edge convention Run `clang++ -std=c++17 -O3 -Wall -Wextra -pedantic markoff_connectivity_exact.cpp -o markoff_exact && ./markoff_exact 1999 && ./markoff_exact 2000 3001` Entry point `Join source_lines with LF, append a terminal LF, and save as markoff_connectivity_exact.cpp` Runtime Apple clang 21.0.0, C++17 standard library, macOS 26.2 arm64 Details For odd primes, the construction uses a complete square-root table and the discriminant of the quadratic in $z$. The binary case uses direct enumeration. Two slots per $(x,y)$ represent the possible roots. The code asserts that every Vieta neighbor is present, checks the total point formula, checks $6(p-1)$ zero-coordinate vertices for $p\equiv1\pmod4$ and zero for $p\equiv3\pmod4$, and checks that component sizes sum to the vertex count. The recorded execution was split at 1999. The two chunks contain 431 prime rows through 3001, including the separate $p=2,3$ audit. Each row gives component sizes, root eccentricity, a farthest vertex, a shortest Vieta word, and self-loop incidences. Every stored word is replayed before the row is emitted. Related: Computation 2 · Computation 3 Open full record →
A second implementation scans all p^3 triples and independently reproduces every component field and shortest-path certificate for the 26 primes through 101.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 3 A second implementation scans all p^3 triples and independently reproduces every component field and shortest-path certificate for the 26 primes through 101. Relevance to this problem For Prime exceptions to connectivity of the Markoff graph, record mgpc-artifact-cubic-oracle (“Independent cubic oracle through p=101”) supplies evidence or a replay used to check the packet. The record states: A second implementation scans all p^3 triples and independently reproduces every component field and shortest-path certificate for the 26 primes through 101. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every prime p with 2 <= p <= 101 under the literal coefficient-one equation and three Vieta-edge convention Run `python3 markoff_connectivity_oracle.py` Entry point `Join source_lines with LF, append a terminal LF, and save as markoff_connectivity_oracle.py` Runtime CPython 3.9.6 standard library, macOS 26.2 arm64 Details This Python oracle shares no surface-construction or state-indexing code with the optimized C++ program. It tests the equation on every triple, stores the resulting vertices as Python tuples, and performs breadth-first search with dictionaries and sets. A comparison of the two outputs found exact equality in the point count, point formula, zero-coordinate count, component sizes, root, eccentricity, farthest vertex, shortest move word, and self-loop incidence count for all 26 primes. Related: Artifact 2 Open full record →
No records match these filters.

Notes and companion material

Original intake status. UNKNOWN as of 2026-07-28. Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. The graph G_p is connected for every prime 5 <= p < 1,000,000 and every prime p > T; this packet also certifies 40,066 primes with 10,000,000 < p <= 20,000,000. The unresolved p >= 5 cases are the primes in 1,000,000 <= p <= T outside the individual certificates recorded or cited here. Literally, G_2 is connected and G_3 has no vertices, so p=3 still needs a null-graph convention. 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: Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. The graph G_p is connected for every prime 5 <= p < 1,000,000 and every prime p > T; this packet also certifies 40,066 primes with 10,000,000 < p <= 20,000,000. The unresolved p >= 5 cases are the primes in 1,000,000 <= p <= T outside the individual certificates recorded or cited here. Literally, G_2 is connected and G_3 has no vertices, so p=3 still needs a null-graph convention.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.

Recorded example 1. Over F_5, (3,3,3) is a vertex and applying the first Vieta involution gives the adjacent vertex (1,3,3).

How the 10 records connect

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

ProblemPrime exceptions to connectivity of the Markoff graph

All 14 recorded relations between these records and the problem

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Square-class collisions in the Pell-Lucas sequencenumber theory

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Prime exceptions to connectivity of the Markoff graph.” TheoremDB. P2738. Problem statement; statement text SHA-256 856b4312bdb4139c8de43c0720842f955bbf8004e58882bfab5a2abd1ae3169d. https://theoremdb.org/statement/?ref=P2738

BibTeX

@misc{theoremdb-problem-856b4312bdb4139c8de43c0720842f955bbf8004e58882bfab5a2abd1ae3169d,
  title = {{Prime exceptions to connectivity of the Markoff graph}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 856b4312bdb4139c8de43c0720842f955bbf8004e58882bfab5a2abd1ae3169d},
  url = {https://theoremdb.org/statement/?ref=P2738}
}

Plain text: Built Markdown snapshot

This problem includes 10 records joined by 14 typed links, current as of July 28, 2026.

1References

The maximal-divisor criterion certifies 40,066 primes between ten and twenty million. Eddy et al., Theorem 1.4; Brown, DOI 10.1007/s40993-024-00592-9, Theorem 2; packet claims mgpc-claim-components-through-3001, mgpc-claim-maximal-divisor-10m-20m, and mgpc-claim-small-characteristics; Theorem 1.5 and Section 7, Data on Connectivity; Self-contained implementation of Theorem 1.5, authored and executed 2026-07-28; Theorems 1.4 and 1.5 and Section 7. ↗preprint · primary source · arXiv:2308.07579v1 · checked 2026-07-28Source use: original summary.Connectivity is proved below one million and beyond an explicit threshold. Let T=(863#)(53#)(13#)(7#)(5#)3^3 2^5. The graph G_p is connected for every prime 5 <= p < 1,000,000 and every prime p > T; this packet also certifies 40,066 primes with 10,000,000 < p <= 20,000,000. The unresolved p >= 5 cases are the primes in 1,000,000 <= p <= T outside the individual certificates recorded or cited here. Literally, G_2 is connected and G_3 has no vertices, so p=3 still needs a null-graph convention. The maximal-divisor criterion certifies 40,066 primes between ten and twenty million. An exhaustive exact-integer scan of all 606,028 primes with 10,000,000 < p <= 20,000,000 proves G_p connected for 40,066 of them by the criterion of Eddy, Fuchs, Litman, Martin, and Tripeny. Exact maximal-divisor criterion scan. Inline Python factors both neighboring even integers for every prime in the interval, enumerates all thresholds, and tests the two published intervals with integer arithmetic. Dated.Also cited at Primary sources checked on 2026-07-27 call universal connectivity conjectural and provide large theoretical and computational ranges, but the audit did not exhaust later literature.Also cited at Theorems 1.4 and 1.5 and Section 7.Also cited at Eddy et al., Theorem 1.4; Brown, DOI 10.1007/s40993-024-00592-9, Theorem 2; packet claims mgpc-claim-components-through-3001, mgpc-claim-maximal-divisor-10m-20m, and mgpc-claim-small-characteristics.Also cited at Theorem 1.5 and Section 7, Data on Connectivity.Primary sources checked on 2026-07-27 call universal connectivity conjectural and provide large theoretical and computational ranges, but the audit did not exhaust later literature.Source used to assess the problem's recorded status.
Matthew de Courcy-Ireland and Seungjae Lee, “Experiments with the Markoff surface”. arXiv:1812.07275 (2018). Independent exact reproduction and extension through p=3001; compare the p<3000 connectivity computation and Proposition 2.1; Introduction, p<3000 computation, and Proposition 2.1. ↗preprint · primary source · arXiv:1812.07275, version checked 2026-07-28 · checked 2026-07-28Source use: original summary.Exact Vieta enumeration connects every G_p for 5 <= p <= 3001. A full breadth-first search visits all 1,169,185,980 nonorigin surface points across all 429 primes with 5 <= p <= 3001 and finds one component at every prime. Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.Also cited at Introduction, p<3000 computation, and Proposition 2.1.Also cited at Independent exact reproduction and extension through p=3001; compare the p<3000 connectivity computation and Proposition 2.1.For Prime exceptions to connectivity of the Markoff graph: Exact Vieta enumeration connects every G_p for 5 <= p <= 3001. A full breadth-first search visits all 1,169,185,980 nonorigin surface points across all 429 primes with 5 <= p <= 3001 and finds one component at every prime. Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.
Colby Austin Brown, “An almost linear time algorithm testing whether the Markoff graph modulo p is connected”. Research in Number Theory 11(1) (2025), 6. DOI 10.1007/s40993-024-00592-9. Brown, Theorem 2 and Sections 1 and 4; source comparison completed 2026-07-28; Theorem 2, definition before Figure 2, and Section 4; Brown, Introduction, comparison with O(p^2) flood fill; packet artifact gives measured small-p execution; Brown, Algorithm 3, Section 4, and the libbgs repository at the pinned commit. ↗ open copy ↗journal article · primary source · version of record · checked 2026-07-28Source use: original summary.Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions. Full-vertex flood fill has quadratic state cost. The route gives complete component and path certificates at small p, while its state count prevents it from crossing the unresolved effective interval. Shard the criterion scan, then route failures to the almost-linear test. Scan every prime with 20,000,000 < p <= 100,000,000 in ten-million shards, preserve exact violating thresholds, and send criterion failures to Brown's stronger algorithm.Also cited at Theorem 2 and Sections 1 and 4.Also cited at The wording and acceptance packet were written by the contributor after checking current primary literature on the Markoff mod-p graph.Also cited at Theorem 2, definition before Figure 2, and Section 4.Also cited at Brown, Introduction, comparison with O(p^2) flood fill; packet artifact gives measured small-p execution.Also cited at Brown, Algorithm 3, Section 4, and the libbgs repository at the pinned commit.Primary sources checked on 2026-07-27 call universal connectivity conjectural and provide large theoretical and computational ranges, but the audit did not exhaust later literature.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Prime exceptions to connectivity of the Markoff graph: The route gives complete component and path certificates at small p, while its state count prevents it from crossing the unresolved effective interval.
Colby Austin Brown, libbgs, Markoff-graph connectivity software, GitHub commit ff9360aa1ed14a35a55511d59a75d8d95fbbdf60 (2026). Repository implementation used to generate the prime connectivity table. ↗software · software source · commit ff9360aa1ed14a35a55511d59a75d8d95fbbdf60 · checked 2026-07-28Source use: original summary.Provides the implementation used for the packet's independent connectivity checks.For Prime exceptions to connectivity of the Markoff graph: Provides the implementation used for the packet's independent connectivity checks.
William Chen, “Nonabelian level structures, Nielsen equivalence, and Markoff triples”. arXiv:2011.12940 (2020). Markoff transitivity corollary. ↗preprint · primary source · arXiv:2011.12940v2 · checked 2026-07-28Source use: original summary.Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.Also cited at Primary sources checked on 2026-07-27 call universal connectivity conjectural and provide large theoretical and computational ranges, but the audit did not exhaust later literature.Primary sources checked on 2026-07-27 call universal connectivity conjectural and provide large theoretical and computational ranges, but the audit did not exhaust later literature.Source used to assess the problem's recorded status.For Prime exceptions to connectivity of the Markoff graph: Primary sources checked on 2026-07-27 call universal connectivity conjectural and provide large theoretical and computational ranges, but the audit did not exhaust later literature.
Daniel E. Martin, “A new proof of Chen's theorem for Markoff graphs”. arXiv:2502.15960 (2025). Abstract and Markoff component-divisibility theorem. ↗preprint · primary source · arXiv:2502.15960v1 · checked 2026-07-28Source use: original summary.Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.For Prime exceptions to connectivity of the Markoff graph: Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.
Elisa Bellah, Claire Dunn, Vernon Naidu, and Alette Wells, “Connectedness of special points in the Markoff mod $p$ graphs”. arXiv:2511.23401 (2025). Abstract and main theorem. ↗preprint · primary source · arXiv:2511.23401v1 · checked 2026-07-28Source use: original summary.Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.For Prime exceptions to connectivity of the Markoff graph: Dated source and convention audit. The checked primary literature still treats universal connectivity as open for p>=5, while proving a finite effective remainder and several broad or bounded regions.
Shohei Satake and Yoshinori Yamasaki, “Topological properties of generalized Markoff mod $p$ graphs”. arXiv:2512.21963 (2025). Abstract; generalized level parameter and topological properties. ↗preprint · primary source · arXiv:2512.21963v1 · checked 2026-07-28Source use: original summary.nearby generalized-level result, not a universal connectivity theorem for the zero levelFor Prime exceptions to connectivity of the Markoff graph: nearby generalized-level result, not a universal connectivity theorem for the zero level

CC0 formulation of the prime-connectivity question with the graph convention written into the statement.

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[#P2738] Prime exceptions to connectivity of the Markoff graph

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