The Collatz Conjecture 🔢

Overview

• Described as the most dangerous problem in mathematics, warned against by senior mathematicians.
• Famous mathematician Paul Erdos commented on it, stating, "Mathematics is not yet ripe enough for such questions."
• Conjecture involves any number eventually ending up in the loop: 4 -> 2 -> 1.
• Known as Collatz conjecture, also Ulam conjecture, Kakutani’s problem, Thwaites conjecture, Hasse’s algorithm, the Syracuse problem, and 3N+1.

How it Works

• Pick a number. If odd, multiply by three and add one. If even, divide by two.
• Example with number 7:
  • 7 (odd) -> 21+1 = 22 (even)
  • 22 -> 11 (odd)
  • 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
  • The loop: 1 -> 4 -> 2 -> 1

Characteristics

• Hailstone numbers: Numbers generated by 3x+1 due to their up-down motion before settling.
• Total stopping time: Total steps to reach the number 1.

Patterns and Analysis

• Randomness in sequences before reaching one, similar to geometric Brownian motion (stock market).
• Distribution of leading digits in sequences follows Benford’s Law, where lower digits (like 1) appear more frequently.
• Despite the growth (3x+1) being powerful, the division by two ensures sequences trend downwards on average.
• Average sequence shrinkage across many iterations.

Visualization

• Directed graph visualization of number sequences.
• Sequences resembling coral or seaweed patterns when visualized with rotations.

Computational Evidence

• Collatz conjecture tested for every positive integer up to 2^68 (approx. 300 quintillion numbers).
• No counterexamples found for finite sequences through brute force testing.

Proof Attempts

• Approaches by Riho Terras, reducing number bounds steadily closer to 1 (1976 -> 1994).
• Terence Tao (2019): Most numbers eventually get arbitrarily small, but not a full proof.

Undecidability

• Collatz conjecture might be inherently undecidable, like the halting problem.
• John Conway showed a generalization of 3x+1 (FRACTRAN) is turing-complete.

Counterexamples and Practical Impacts

• Experience with negative numbers showing multiple loops.
• Real-world detection examples using Benford’s Law in fraud detection and elections.
• Open questions: Growth off to infinity? Unique counterexamples? Sequences in unexplored parts?

Summary

• Collatz conjecture remains a mystery, deeply engaging yet unsolved, prompting deep mathematical contemplation and appreciation for numerical patterns and complexities.

Final Thoughts

• Illustrates the unpredictable and peculiar nature of numbers.
• Demonstrates that despite exhaustive searches and sophisticated analysis, simple questions can remain intractable.