The Collatz Engine
Exploring one of mathematics' most famous unsolved problems.
The engine continuously catalogs, visualizes, and analyzes Collatz trajectories using persistent computation, transparent records, and live mathematical telemetry.
This project does not claim to prove the Collatz Conjecture. It is a public exploration and visualization system.
Engine Status
OFFLINE
Live catalog active
Runtime
Not started
Waiting for engine start
Current Number
1
Next integer scheduled
Numbers Checked
0
Highest: 0
Highest Peak
Pending
Largest value encountered
Longest Trajectory
0 steps
Current record trajectory length
Throughput
Pending
Awaiting next run
Your Local Time
Pending
Detecting timezone...
Collatz Trajectory Visualizer
n=27 · 111 steps · peak 9,232 · log scale
Engine is not live-estimating; showing the latest backend-verified trajectory.
Estimated Live using latest verified n=27 · 111 steps · peak 9,232
Estimated Live Sequence Trace
Generated locally from n. Not catalog-verified until backend sync.
Start Number
27
Steps to 1
111
Peak Value
9,232
Odd Step Density
36.9%
| Step | Value | Type | Operation | Result |
|---|---|---|---|---|
| 001 | 27 | Odd | 3n + 1 | 82 |
| 002 | 82 | Even | n / 2 | 41 |
| 003 | 41 | Odd | 3n + 1 | 124 |
| 004 | 124 | Even | n / 2 | 62 |
| 005 | 62 | Even | n / 2 | 31 |
| 006 | 31 | Odd | 3n + 1 | 94 |
| 007 | 94 | Even | n / 2 | 47 |
| 008 | 47 | Odd | 3n + 1 | 142 |
| 009 | 142 | Even | n / 2 | 71 |
| 010 | 71 | Odd | 3n + 1 | 214 |
Showing 10 of 111 steps · Estimated Live using latest verified n=27 · local visualization only; catalog verification arrives on backend sync
Estimated Live Descent Profile
Generated locally from the estimated engine position.
Engine is not live-estimating; showing the latest backend-verified trajectory.
Start
27
Peak
9,232
First descent
Step 96
Total steps
111
Peak / Start
341.93×
Estimated Live Peak Window
Browser-generated preview around the estimated engine position. Not an official catalog record.
Highest Peak in Preview
Pending
Produced by n ≈
Pending
Numbers Previewed
0
Peak preview data will appear once the selected starting window is available.
Estimated Live Stopping-Time Window
Browser-generated preview around the estimated engine position. Not an official catalog record.
Longest trajectory in preview
Pending
Produced by n ≈
Pending
Peak for record n
Pending
Numbers previewed
0
Stopping-time preview data will appear once the selected starting window is available.
Estimated Live Odd/Even Transition Graph
Generated locally from the estimated engine position.
Odd transitions
41
Even transitions
70
Odd / Even ratio
0.59
Longest odd run
1
Longest even run
5
All-Time Engine Records
AuthoritativeAuthoritative records preserved in engine state and permanent record storage.
Longest Trajectory
Pending
Starting n: not retained
Highest Peak
Pending
Starting n: not retained
Numbers Checked
Pending
Engine state unavailable
Current Number
Pending
Engine state unavailable
Last Updated
Pending
Engine state timestamp
Top 10 All-Time Longest Trajectories
| Rank | Starting Number (n) | Steps | Peak Value | Discovered | Source |
|---|---|---|---|---|---|
| No permanent records preserved yet. | |||||
Detailed starting number for this historical record was not retained. A reconstruction backfill can restore it.
Top 10 All-Time Highest Peaks
| Rank | Starting Number (n) | Peak Value | Steps | Discovered | Source |
|---|---|---|---|---|---|
| No permanent records preserved yet. | |||||
Detailed starting number for this historical record was not retained. A reconstruction backfill can restore it.
Historical reconstruction not started. Permanent rows currently reflect retained data and future live preservation. Missing historical starting numbers are not inferred.
Backend Record Leaders
Official backend records from the cached dashboard payload. Estimated previews stay separate.
Longest Trajectories (Backend Records)
| Rank | Starting Number (n) | Steps | Peak |
|---|---|---|---|
| No persisted records available yet. | |||
Highest Peaks (Backend Records)
| Rank | Starting Number (n) | Peak Value | Steps |
|---|---|---|---|
| No persisted records available yet. | |||
These tables reflect backend-verified dashboard records. Estimated live previews never change official records.
This engine records computational observations only. It does not claim to prove the Collatz Conjecture.
Key Records
Awaiting dataset growth · Records update automatically as the engine catalogs numbers
Longest Path
Pending
Awaiting dataset growth
Highest Peak
Pending
Awaiting dataset growth
Highest Peak Ratio
Pending
Awaiting dataset growth
Numbers Cataloged
Pending
Engine state unavailable
Highest n Checked
Pending
Engine state unavailable
Catalog Status
Pending
Not yet started
Records will appear here as the engine catalogs trajectories. All verified numbers reach 1.
Milestone Feed
Tracking verified catalog growth as the autonomous engine advances sequentially from 1 upward.
Milestones are computation progress markers, not mathematical proof markers.
Verified Catalog Size
Pending
Previous Milestone Reached
First milestone pending
Current Target
1M cataloged
1,000,000 numbers remaining
ETA
Calculating
Collecting sustained-rate baseline
Progress Toward Current Target
0.00%
Next Upcoming Milestones
System Integrity
The engine processes integers sequentially in verified batches. Completed results are stored in the catalog and displayed here as a live computational record. This system does not claim to prove the Collatz Conjecture.
Live checks monitor recent catalog health. Full verification scans review the catalog for duplicate entries, missing ranges, and record consistency.
Limited export samples are capped for public access.
Live Bounded Check
Fast, current, and limited to the latest catalog window for public dashboard health.
Highest Verified n
Pending
Numbers Cataloged
Pending
Last Live Check
Pending
Duplicate Check
Pending
Missing Range Check
Pending
Record Consistency
Pending
Worker Heartbeat
Pending
Status Readable
Pending
Full Catalog Verification
Last persisted full scan across the catalog, including duplicate entries, missing ranges, and record consistency.
Live summary covers the most recent Pending catalog entries.
How the Engine Works
The Collatz Engine evaluates each integer in order, records the trajectory statistics, and updates this dashboard from the verified catalog. Batching improves performance without skipping integers. This project is a public computational exploration, not a proof.
Sequential coverage
The engine starts at 1 and advances one integer at a time, preserving a clear verified range.
Trajectory evaluation
Each integer is evaluated with the Collatz rule until the sequence reaches 1.
Verified batches
Results are written in batches for efficiency while maintaining sequential coverage.
Live catalog
The dashboard reads from the verified catalog and updates as new batches complete.
Integrity checks
Checks monitor duplicate entries, missing ranges, and record consistency.
Computational scope
This is computational exploration, not a proof of the conjecture.
Limited export samples are capped for public access.
AI Observatory Notes
AI-assisted summaries generated from verified Collatz Engine data. Notes require human review before publication and do not claim to prove the conjecture.
Infrastructure Note: Engine Integrity Report — 2026-06-02
As of data captured on 2026-06-02, the Collatz Engine Observatory reports the engine status as running. The current processing position is 8.974M, matching the total count of 8.974M numbers verified.…
Read noteWeekly Digest: Collatz Engine Progress — 2026-06-01
The Collatz Engine Observatory continues to run its computational verification workflow for the Collatz process, tracking each tested starting value and recording trajectory behavior observed during …
Theoretical Lens: Mathematical Perspectives on Collatz Trajectories
One useful way to examine the Collatz process is through stopping times: the number of steps required for a starting value to fall below its initial value, or to reach 1. This lens does not attempt t…
Weekly Digest: Collatz Engine Progress — 2026-06-02
The Collatz Engine Observatory continues its computational verification run, with the engine currently reporting an active running status as of the latest captured dataset on 2026-06-02. The current …
New Longest Trajectory: 685 Steps Reached
# New Longest Trajectory: 685 Steps Reached
AI notes summarize verified engine data. They do not constitute a proof of the Collatz Conjecture.
Heatmaps & Pattern Views
Latest 200 verified trajectories · n = pending · refreshed not yet refreshed · refresh cadence: 5 min
Highlights where longer trajectories appear inside the latest verified window.
Rows = bucketed trajectory groups · Columns = cataloged number ranges · Color = relative activity intensity
Estimated Live Near-Escape Candidates
Visualization OnlyGenerated locally from the estimated engine position. These are live visualization candidates, not verified catalog records.
Browser-computed window around estimated n=~...
What is a near-escape candidate?
Visualization label onlyNear-escape candidates are trajectories that climb unusually high, delay descent, or show unusually high odd-step density before collapsing back to 1. These numbers are flagged for analytical interest based on configurable thresholds. All verified numbers reach 1. Near-escape is a visualization label, not a mathematical claim.
Waiting for estimated position
Browser-generated candidates will appear as soon as a valid visualization source is available.
Odd step (3n+1)
Waiting for estimated engine position. No database query is being made for this panel.
Even step (n/2)
The Collatz Conjecture
A plain-language guide to one of mathematics' most intriguing open problems
What is the Collatz Conjecture?
The Collatz Conjecture is a problem in mathematics that concerns a sequence defined by two rules: if a number is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat this process with the result.
Definition of the Collatz Function
Let n be a positive integer. Define f(n) as: n/2 if n is even, or 3n+1 if n is odd. The conjecture states that for every positive integer n, repeated application of f eventually reaches 1.
Behavior of Collatz Sequences
Collatz sequences exhibit a fascinating mix of seemingly chaotic rises and steady descents. Most trajectories peak early and then decline irregularly toward 1. Despite extensive computation, no counterexample has ever been found.
How The Collatz Engine Works
The Collatz Engine uses computation to catalog, analyze, and visualize Collatz sequences for positive integers as the verified catalog grows. It surfaces unusual trajectories and records without claiming proof.
FAQs
Is the Collatz Conjecture proven? What is the largest number tested? How are records determined? Can I contribute? Where can I see more data? All questions answered in our FAQ section.
Data & Methodology
What this engine computes, stores, and observes, and what it does not claim
What the engine catalogs
- The full trajectory sequence for each starting number n
- Total step count (length of path to 1)
- Peak value reached and the step at which it occurs
- Peak ratio (peak / n) as a normalized comparison metric
- First descent delay: steps before the value first falls below n
- Odd-step count and odd-step density (odd steps / total steps)
What data is stored
- Compact batch summaries: max steps, peak values, ratios, and residue stats, not full sequences for every number
- Full sequences are reserved for record breakers, near-escape candidates, and selected demo samples
- Batch summary data is local until persistence is added in Phase 5
- Record-breaking trajectories are flagged and stored in full for inspection
- AI-drafted observation notes remain private until admin-approved
- No personally identifiable data is collected from visitors
AI-assisted observations
- An AI model reviews batches of trajectory statistics for statistical patterns
- Observations are generated as private draft notes and never published automatically
- Every observation requires explicit admin approval before appearing publicly
- Notes describe statistical observations only. No proof claims are made
- The AI model does not have access to unpublished drafts from other sessions
Why no proof claim
- The Collatz Conjecture is an open problem in mathematics. No proof or disproof is known
- This engine verifies that specific numbers reach 1; it does not prove the general case
- Verification for n ≤ N tells us nothing about all n > N without a mathematical proof
- AI-drafted notes are statistical observations, not mathematical arguments
- This project makes no claim to be advancing a proof of the conjecture
Prior Work & Related Projects
This engine is not the largest or fastest verification effort. Here is what it is
Scope clarification
Extensive Collatz verification efforts exist. The verified range already exceeds 2⁶⁸. This engine is not attempting to surpass those records. Its purpose is public visualization, systematic cataloging of trajectory statistics, and human-reviewed AI observations over a tractable range.
Public visualization
The primary goal is a visually accessible interface for exploring Collatz trajectories, browsable by starting number, sortable by metric, and readable without a mathematics background.
Trajectory cataloging
The engine systematically computes and stores per-trajectory statistics for a defined range, making the data queryable and comparable in ways a one-off script does not.
Pattern exploration
Heatmaps, distribution charts, and record leaderboards surface statistical structure across large ranges. These patterns are visible only in aggregate.
Human-reviewed AI observations
AI-assisted analysis generates private draft notes that require admin approval before publication. This creates a reviewed, citable layer of statistical commentary without making proof claims.
Notable prior & related efforts
Oliveira e Silva verification
Verified all n up to 2⁶⁸ (≈ 2.95 × 10²⁰)
The most extensive verified range as of this writing. This engine does not approach that scale.
BOINC / distributed computing efforts
Crowd-sourced verification across volunteer computers
This engine runs on a single server; it is not a distributed effort.
OEIS sequence A006577
Step-count sequence for all positive integers, catalogued in the OEIS
A definitive reference for step counts; this engine independently computes the same values as verification.
Lagarias (2010) annotated bibliography
Comprehensive survey of Collatz-related literature
The authoritative academic reference for the conjecture's history and open problems.
Get Involved
Submit an observation, report an issue, share an idea, or contact the project directly.
About This Project
The Collatz Engine is an independent public autonomous mathematics exploration system created by Amaete Umanah. It catalogs behavior, shares verified computational results, and does not claim to prove the Collatz Conjecture.
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Support the Engine
The Collatz Engine is a public autonomous mathematics exploration system. Contributions help support compute, hosting, data exports, and continued development.
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Use this form to submit an observation, report an issue, suggest an improvement, or contact the project.