Solving the Collatz Conjecture with SymPy

Discover how to implement the `Collatz Conjecture` in SymPy with a clear and practical guideline. This post also reviews symbolic computation techniques in Python.
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Solving the Collatz Conjecture with SymPy: A Step-by-Step Guide

The Collatz Conjecture is a simple yet fascinating mathematical problem that has perplexed mathematicians for decades. It involves a process of iterative sequences based on a starting natural number and the operations defined by whether the number is even or odd. In this guide, we’ll explore how to express the Collatz Conjecture using the SymPy library in Python.

The Problem at Hand

The goal is to create a function in SymPy that takes a natural number n and returns the number of steps it takes to reach 1 using the Collatz procedure. The classic Collatz function is defined as follows:

If n is even, divide it by 2.

If n is odd, multiply it by 3 and then add 1.

The process continues until n reaches 1.

For example, for n = 27, the sequence is as follows:

[[See Video to Reveal this Text or Code Snippet]]

The total steps for reaching 1 in this example are 111.

Implementing the Collatz Conjecture in Python

Here's the typical implementation of the Collatz function in Python:

[[See Video to Reveal this Text or Code Snippet]]

This function works perfectly for inputting integer values. However, using SymPy for symbolic computation presents a new challenge.

Translating the Algorithm to SymPy

To implement the Collatz function in SymPy, you can follow these steps:

Step 1: Define the Function

First, we need to declare a symbolic function that represents the output of the Collatz sequence:

[[See Video to Reveal this Text or Code Snippet]]

Step 2: Create the Equation

Next, we will set up our equation, substituting the input for a symbolic variable:

[[See Video to Reveal this Text or Code Snippet]]

Step 3: Implement the Substitution and Replacement

Now, we need to substitute our input with an actual integer and replace any non-symbolic results:

[[See Video to Reveal this Text or Code Snippet]]

Step 4: Replace Non-symbolic Results

We can use the replace method to convert symbolic results into their corresponding Collatz output:

[[See Video to Reveal this Text or Code Snippet]]

Automatic Evaluation

If you desire automatic evaluation, consider creating a SymPy class that derives from Function with an eval method to check for integer inputs. This follows a similar structure to existing SymPy functions, like cos.

Conclusion

The Collatz Conjecture offers a rich area for experimentation in mathematical programming, and with SymPy, we can both explore and evaluate these sequences symbolically. Whether you're a seasoned programmer or just getting started with Python, understanding and implementing mathematical conjectures can be deeply rewarding. Happy coding!

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