Collatz Conjecture Calculator – 3x+1 Sequence Explorer
Explore the Collatz conjecture (3x+1 problem) with an interactive calculator that showcases hailstone sequences, stopping times, and orbit peaks for any positive starting integer.
Enter any positive integer ≥ 1. Classic examples: 6, 7, 19, 27, 871.
Safety cap prevents runaway sequences; raise it for deeper dives.
Switch to log scale for large peaks.
Trajectory chart
Enter values to render the orbit.
Sequence table (orbit)
Each row shows one step of the iteration.
| Step | Value | Parity |
|---|---|---|
| Sequence data will appear here after you generate a trajectory. | ||
How to use this Collatz explorer responsibly
Use the controls to simulate hailstone sequences, inspect stopping-time behavior, and compare peaks across starting values.
- Teaching and learning about iteration, orbits, and discrete dynamical systems.
- Gaining intuition for stopping times, convergence speed, and peak values.
- Exploratory experimentation in number theory and computer science education.
For serious research, rely on dedicated high-performance implementations that handle extremely large starting values with rigorous record-keeping and numerical safeguards.
Methodology
This calculator steps through the 3x+1 rule (even numbers halve, odd numbers triple-plus-one) while tracking stopping times, total stopping time, and the orbit peak. The simulation halts when the sequence reaches 1 or the iteration cap is hit.
Full original guide (expanded)
The Collatz conjecture remains one of the most famous unsolved problems in elementary number theory. Despite a deceptively simple rule, it resists proof. Lothar Collatz introduced the 3x+1 iteration in 1937, and computers have since verified it for numbers up to over 268, yet no general proof is known.
In the words of Jeffrey Lagarias, the problem is “extraordinarily difficult” and “completely out of reach of present-day mathematics.”
References & further reading
- Wikipedia, “Collatz conjecture” – formal definition, history, and computational records.
- Quanta Magazine, “The Simple Math Problem We Still Can’t Solve” – accessible overview of recent work.
- Jeffrey C. Lagarias, “The 3x+1 Problem and Its Generalizations” – research-level survey of the 3x+1 problem.
More in Core Math & Algebra
Suggested explorations
Try these sequences and compare the stopping times and peaks.
- n = 7 – short orbit with a moderate peak.
- n = 27 – famous for its surprisingly long trajectory.
- n = 97 – long total stopping time compared with smaller numbers.
- n = 871 – classic example in many Collatz discussions.
For classroom use, compare stopping times and peak values across these examples.