What is Collatz Conjecture?

Use this calculator to compute Collatz Conjecture values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Collatz Conjecture used?

Sequences, calculus, physics, engineering, and related disciplines.

What formulas does this use?

All standard collatz conjecture formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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MATHEMATICSSequencesMathematics Calculator

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Collatz Conjecture

Explore the 3n+1 problem and calculate Collatz sequences for any starting value. Visualize trajectories, odd/even step distribution, and more.

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Why: Understanding collatz conjecture helps you make better, data-driven decisions.

How: Enter Starting Value (n) to calculate results.

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MATHEMATICS

The 3n+1 Problem — Mathematics' Simplest Unsolved Mystery

Pick any positive integer. If even, halve it. If odd, triple it and add 1. Repeat. Does every sequence eventually reach 1? Nobody knows.

🔢 Sample Examples — Click to Load

Starting Number

Starting Value (n)

Use logarithmic scale for trajectory chart (recommended for large values)

For educational and informational purposes only. Verify with a qualified professional.

📋 Key Takeaways

• Unsolved since 1937 — Despite its simplicity, no proof exists
• Simple rules: If even → n/2; if odd → 3n+1. Repeat until 1
• Always reaches 1 (conjectured) — Verified for all n up to 2⁶⁸
• $1M prize unclaimed — Part of unsolved mathematics challenges

💡 Did You Know?

📜Paul Erdős said: 'Mathematics is not ready for such problems' — the Collatz conjecture remains stubbornly unprovenSource: Erdős

🔢The conjecture has been verified computationally for all starting values up to 2^68 (≈295 quintillion)Source: Computational verification

👤Lothar Collatz proposed the problem in 1937 while a student at the University of HamburgSource: History

🌀The sequence exhibits chaotic behavior — small changes in input can cause wildly different trajectoriesSource: Chaos theory

❄️Numbers are called 'hailstone numbers' because values rise and fall like hailstones in a stormSource: Terminology

🤖AI and machine learning have been used to search for patterns, but no proof has emergedSource: Modern research

📖 How the Collatz Conjecture Works

Start with any positive integer n. Apply these rules repeatedly:

The Rules

If n is even: divide by 2 → n/2
If n is odd: multiply by 3 and add 1 → 3n+1

Example: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

🎯 Expert Tips

Patterns in Stopping Times

Stopping times often cluster — numbers with similar binary representations may behave similarly.

Powers of 2 Are Trivial

2, 4, 8, 16, 32... reach 1 in log₂(n) steps by repeated halving.

Number Theory Connection

The conjecture relates to Diophantine equations, ergodic theory, and dynamical systems.

Visualization Techniques

Use log scale for large sequences; odd/even step distribution reveals structure.

⚖️ Comparison: This Calculator vs Manual vs Programming

Feature	This Calculator	Manual	Programming
Instant results	✅	❌ Slow	✅
Visual charts	✅	❌	⚠️ Custom
Odd/even breakdown	✅	❌	✅
Large numbers (9B+)	✅	❌	✅
Share & copy	✅	❌	❌
Educational content	✅	❌	❌

❓ Frequently Asked Questions

Is the Collatz conjecture proven?

No. Despite being verified for astronomically large numbers (up to 2^68), no general proof exists. It remains one of mathematics' most famous open problems.

Why is the Collatz conjecture so hard to prove?

The rules are simple, but the behavior is chaotic. Proving that every sequence eventually reaches 1 requires showing that no infinite cycle or divergent trajectory exists — and that has eluded mathematicians for nearly 90 years.

What is the 'stopping time'?

The stopping time is the number of steps required for the sequence to reach 1 for the first time. For example, 27 has a stopping time of 111.

What is the largest number ever tested?

Computational verification has reached 2^68 (about 295 quintillion). Every number below that has been confirmed to eventually reach 1.

Who proposed the Collatz conjecture?

German mathematician Lothar Collatz proposed it in 1937. It has also been attributed to others and is sometimes called the 3n+1 problem or hailstone sequence.

Are there connections to other mathematics?

Yes. The conjecture connects to number theory, dynamical systems, ergodic theory, and even has implications for computer science and cryptography.

What is the record for longest stopping time?

The number 9,780,657,631 holds the record for the highest known total stopping time among numbers below 10^10, with 1,132 steps.

Is there a prize for solving it?

While not an official Millennium Prize, various organizations have offered rewards. The Clay Mathematics Institute focuses on other problems, but the Collatz conjecture remains a celebrated open problem.