Collatz Conjecture Calculator

What is the Collatz Conjecture?

The Collatz Conjecture is a mathematical problem that proposes a sequence of steps for any positive integer. The conjecture states that when the following rules are applied, the sequence will eventually reach the number 1:

• If the number is even, divide it by 2.
• If the number is odd, multiply it by 3 and add 1.

For example, starting with the number 6, the sequence is:

\[ 6 \to 3 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1 \]

The conjecture remains unproven, but it has been verified for a vast range of numbers. It’s often used as an example to illustrate the beauty and unpredictability of simple mathematical rules.

Formula for the Collatz Conjecture

The sequence for the Collatz Conjecture can be written as:

\[ f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \\ 3n + 1, & \text{if } n \text{ is odd} \end{cases} \]

Purpose of the Collatz Conjecture Calculator

This calculator allows users to explore the Collatz Conjecture interactively. You can input any positive integer to generate its Collatz sequence and view the step-by-step calculations. Additionally, the calculator provides an option to define custom rules for even and odd numbers, offering a fun way to experiment with variations of the conjecture.

How to Use the Calculator

Follow these simple steps to use the calculator effectively:

• Enter a positive integer in the input field.
• Select one of the two options:
  • Use Default Rules: Applies the standard Collatz rules.
  • Enter Custom Rules: Define your own formulas for even and odd numbers.
• If using custom rules, enter valid mathematical expressions (e.g., \( n / 2 \) for even and \( 3 \times n + 1 \) for odd).
• Click the Generate button to calculate the sequence and view the step-by-step explanation.
• Click the Clear button to reset the input and start a new calculation.

Features of the Calculator

• Interactive Exploration: Enter any positive integer to generate its sequence.
• Custom Rules: Experiment with your own formulas for even and odd numbers.
• Step-by-Step Details: View how each step of the sequence is calculated.
• Formatted Output: Results and steps are displayed using clean mathematical notation.

FAQs

1. What is the maximum number of steps the calculator can generate?

The calculator limits the sequence to 1,000 steps to prevent excessively long calculations for very large numbers or complex custom rules.

2. Can I use custom rules that involve more complex formulas?

Yes! You can use any valid mathematical expression as a custom rule, such as \( n^2 + 1 \) for odd numbers or \( n / 3 \) for even numbers. Just ensure the rules make sense for integer values.

3. What happens if I enter invalid custom rules?

The calculator will alert you if your custom rules contain invalid mathematical expressions. Double-check your formulas and try again.

4. Is the Collatz Conjecture proven?

No, the Collatz Conjecture remains unproven. It has been verified for a wide range of numbers, but a general proof has not been found.

Conclusion

The Collatz Conjecture Calculator is a fun and educational tool that brings a classic mathematical problem to life. Whether you’re exploring standard rules or creating your own, this calculator provides a hands-on way to learn about sequences and mathematical logic. Give it a try and see where the sequence takes you!