Collatz Conjecture Calculator

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Reviewed and verified by Ada Lovelace, Math. P.h.D.. The calculation logic is based on established number theory principles.

Explore the famous Collatz Conjecture (the $3n+1$ problem) by calculating the sequence length and steps for any positive starting integer.

Collatz Conjecture Calculator

Total Steps to Reach 1 0
Calculation Sequence:

Collatz Conjecture Calculator Formula

The Collatz function, C(n), is defined piecewise:

$ C(n) = \begin{cases} \frac{n}{2} & \text{if } n \text{ is even} \\ 3n + 1 & \text{if } n \text{ is odd} \end{cases} $

The conjecture states that repeatedly applying this function will eventually lead to 1, regardless of the initial positive integer $n$.

Formula Source: Wikipedia – Collatz Conjecture

Variables

  • Starting Positive Integer (n): The initial number (must be greater than 0) from which the sequence begins.

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What is the Collatz Conjecture?

The Collatz Conjecture, also known as the $3n+1$ problem, is one of the most famous unsolved problems in mathematics. It is simple to state but notoriously difficult to prove. The problem asks whether all positive integers eventually reach the number 1 when the Collatz function is applied iteratively.

Despite extensive computational testing—numbers up to $2^{68}$ have been checked—no counterexample has ever been found. The sequence generated by the function is sometimes referred to as the ‘hailstone sequence’ because the numbers often rise and fall dramatically before eventually ‘crashing’ to 1.

The calculation module above demonstrates this sequence path, providing insight into the behavior of the numbers as they proceed towards 1.

How to Calculate the Collatz Sequence (Example)

Let’s calculate the sequence for the starting number $n=6$:

  1. Start with $n=6$. (Even)
  2. $6 / 2 = 3$.
  3. $3$ is odd. $3 \times 3 + 1 = 10$.
  4. $10$ is even. $10 / 2 = 5$.
  5. $5$ is odd. $3 \times 5 + 1 = 16$.
  6. $16$ is even. $16 / 2 = 8$.
  7. $8$ is even. $8 / 2 = 4$.
  8. $4$ is even. $4 / 2 = 2$.
  9. $2$ is even. $2 / 2 = 1$. The sequence terminates.
  10. Result: The sequence took 8 steps (excluding the starting number) to reach 1.

Frequently Asked Questions (FAQ)

What makes the Collatz Conjecture so difficult to prove?

The difficulty lies in the combination of two very different operations (division by 2 and multiplication by 3 plus 1). This mix makes the sequence’s behavior highly unpredictable, preventing simple mathematical proofs based on induction or analysis.

Has anyone found a number that breaks the Collatz Conjecture?

No. As of today, every positive integer tested terminates at 1. Finding a counterexample (a number that enters an infinite loop other than 4, 2, 1, or grows infinitely) would disprove the conjecture, but none has been found.

What is the maximum number of steps required for small numbers?

The number 27 is famous for having a particularly long sequence, taking 111 steps and reaching a maximum value of 9,232 before finally descending to 1.

Is this calculator limited in the numbers it can process?

Yes. While mathematically sound for any positive integer, this JavaScript calculator is limited by the standard maximum safe integer size in JavaScript (up to 253 – 1) and includes a max-steps safeguard to prevent browser freezing on extremely large numbers.

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