Collatz Conjecture Calculator

Welcome to the Collatz Conjecture Calculator, an interactive tool for exploring one of the most fascinating unsolved problems in mathematics. Enter any positive integer and watch how the hailstone sequence unfolds through a series of simple rules until it inevitably reaches the 4 → 2 → 1 loop. The interactive trajectory chart, step-by-step breakdown, and comprehensive statistics help you visualize and understand the surprising behavior of the Collatz sequence.

What is the Collatz Conjecture?

The Collatz conjecture, also known as the 3n+1 problem, the Syracuse problem, or the hailstone problem, is one of the most famous unsolved problems in mathematics. It was first proposed by German mathematician Lothar Collatz in 1937.

The conjecture states: Start with any positive integer n. If n is even, divide it by 2. If n is odd, multiply by 3 and add 1. Repeat this process. The conjecture asserts that no matter what starting number you choose, the sequence will always eventually reach 1.

The Collatz Rules

Starting from any positive integer \(n\), repeatedly applying \(f\) produces a sequence called the hailstone sequence (or Collatz sequence). The conjecture claims this sequence always reaches 1, after which it enters the cycle 1 → 4 → 2 → 1.

Why is It Called the Hailstone Sequence?

The sequence is called a hailstone sequence because the values rise and fall erratically, much like a hailstone that gets blown up and down inside a storm cloud before eventually falling to the ground. When an odd number is tripled and incremented, the value shoots up; when even numbers are halved, the value drops back down. Eventually, the "hailstone" reaches the ground — the number 1.

How to Use This Calculator

1. Enter a starting number: Type any positive integer into the input field. Try the quick examples for famous starting values like 27 or 871.
2. Generate the sequence: Click "Generate Sequence" to compute the full hailstone sequence.
3. Explore the trajectory: The interactive chart shows the value at each step. Toggle between linear and log scale for better visualization of extreme peaks.
4. Review statistics: Check stopping time, peak value, growth ratio, and even/odd step counts.
5. Study step-by-step: The detailed table shows every operation applied at each step, with color coding for even (n/2) and odd (3n+1) steps.

Understanding the Results

Key Statistics

• Stopping Time: The total number of steps to reach 1. Also called the total stopping time.
• Peak Value: The highest number reached during the sequence. Can be surprisingly large even for small starting values.
• Growth Ratio: The ratio of peak value to starting value. Shows how much the sequence "grows" before descending.
• Even Steps: Number of times n/2 was applied (values that were even).
• Odd Steps: Number of times 3n+1 was applied (values that were odd).

Sequence Trajectory Chart

The interactive chart visualizes the hailstone sequence with three highlighted points:

• Green dot — Starting value
• Red dot — Peak value (highest point)
• Gold dot — Final value (1)

For sequences with very large peaks, switch to log scale to see the overall shape more clearly.

Famous Examples

The Number 27

The number 27 is perhaps the most famous starting value in Collatz conjecture research. Despite being a small number, it produces a sequence of 111 steps and reaches a peak of 9,232 — over 341 times its starting value. This dramatic behavior makes it a classic example of the conjecture's unpredictability.

Record Holders for Longest Sequences

Mathematical Properties

Even vs. Odd Step Ratio

In a typical Collatz sequence, even steps (n/2) significantly outnumber odd steps (3n+1). This is because each odd step produces an even number (3n+1 is always even when n is odd), which is then immediately halved. On average, the ratio of even to odd steps is roughly 2:1, which is one heuristic argument for why sequences tend to decrease overall.

The 4-2-1 Loop

Every Collatz sequence that reaches 1 then enters the cycle: 1 → 4 → 2 → 1. The conjecture can be equivalently stated as: "There is no other cycle," meaning no starting number enters a cycle that does not include 1, and no sequence diverges to infinity.

Computational Verification

The Collatz conjecture has been computationally verified for all starting values up to approximately \(2.95 \times 10^{20}\) (as of 2020). While this is strong evidence, it does not constitute a proof.

History and Notable Research

• 1937: Lothar Collatz first posed the conjecture while studying at the University of Hamburg.
• 1970s: The problem gained wide attention in the mathematical community and acquired many names (Syracuse, Ulam, Kakutani).
• 1985: Jeffrey Lagarias published a comprehensive survey and showed connections to number theory and dynamical systems.
• 2019: Terence Tao proved that "almost all" Collatz orbits attain almost bounded values, the strongest partial result toward the conjecture to date.

Paul Erdős famously said about the Collatz conjecture: "Mathematics may not be ready for such problems."

Frequently Asked Questions

What is the Collatz conjecture?

The Collatz conjecture (also known as the 3n+1 problem) states that for any positive integer, if you repeatedly apply the rule "if even, divide by 2; if odd, multiply by 3 and add 1", the sequence will always eventually reach 1. Despite its simple rules, this conjecture remains unproven since Lothar Collatz first proposed it in 1937.

What is a hailstone sequence?

A hailstone sequence (also called a Collatz sequence) is the series of numbers produced by repeatedly applying the Collatz rules to a starting number until reaching 1. It is called a "hailstone" sequence because the values go up and down like a hailstone in a cloud before eventually falling to the ground (reaching 1).

What is the stopping time in the Collatz conjecture?

The stopping time (or total stopping time) is the number of steps it takes for a starting number to reach 1 in its Collatz sequence. For example, starting from 27, the stopping time is 111 steps. The stopping time varies greatly between different starting numbers and does not follow a simple pattern.

Why is 27 a famous number in the Collatz conjecture?

The number 27 is famous in Collatz conjecture research because despite being relatively small, it produces a surprisingly long sequence of 111 steps and reaches a peak value of 9,232 — over 341 times its starting value. This makes it a classic example of how unpredictable the Collatz sequence can be.

Has the Collatz conjecture been proven?

No, the Collatz conjecture has not been proven as of 2024. It has been verified computationally for all starting values up to approximately \(2.95 \times 10^{20}\), but a general mathematical proof remains elusive. In 2019, Terence Tao proved that the conjecture is true for "almost all" numbers in a measure-theoretic sense.

What is the longest Collatz sequence for small numbers?

Among numbers under 1,000, the number 871 has the longest Collatz sequence at 178 steps. Under 10,000 it is 6,171 with 261 steps. Under 100,000 it is 77,031 with 350 steps. Under 1,000,000, the record holder is 837,799 with 524 steps.

Additional Resources

• Collatz Conjecture - Wikipedia
• Hailstone Sequence - Wikipedia