The Collatz Conjecture

Overview

• Considered one of the most dangerous problems in mathematics.
• Simple conjecture; unresolved by even the best mathematicians.
• Paul Erdős stated, "Mathematics is not yet ripe enough for such questions."

The Rules

1. If the number is odd: Multiply by 3 and add 1.
2. If the number is even: Divide by 2.

Example with Number 7

• Start with 7 (Odd):
  • 3 * 7 + 1 = 22 (Even)
  • 22 / 2 = 11 (Odd)
  • 11 (Odd): 3 * 11 + 1 = 34 (Even)
  • Continue until reaching 1: 4 -> 2 -> 1 (Loop)

The Conjecture

• Every positive integer will eventually reach the 4-2-1 loop.
• Also known as the Ulam conjecture, Kakutani's problem, Thwaites conjecture, and 3n + 1.

Terminology

• Hailstone numbers: Numbers generated through the process, due to their rising and falling behaviors.
• Total stopping time: Number of steps to reach 1.

Notable Numbers

• Number 27 reaches as high as 9,232 before reaching 1 (111 steps).
• Paths taken vary widely among adjacent numbers.

Challenges and Research

• Mathematicians struggle to find patterns or make progress.
• Jeffrey Ligarius advises against working on this conjecture for a career.
• Alex Kontorovich and Yakov Sinai investigate the paths of numbers for patterns.

Statistical Analysis

• Paths show randomness resembling geometric Brownian motion.
• Hailstone sequences analyzed for their leading digits show a stable pattern consistent with Benford's Law.
• Benford's Law applies to various natural phenomena, indicating a universal distribution of leading digits.

Proof Attempts

• No proof of all numbers falling into the loop found yet.
• Counterexamples: A number or sequence that does not comply would disprove the conjecture.
• No loop or number leading to infinity has been found despite testing up to 2^68.
• Possible existence of a counterexample remains a theoretical concern.

Mathematical Insights

• Odd numbers: When multiplied and then divided, they tend to shrink.
• Average growth factor of odd numbers in sequences is less than 1 (3/4 on average).
• Visualizing sequences via directed graphs helps illustrate connections.

Research Findings

• Terry Tao suggests almost all numbers will reach smaller values in their sequences.
• Patterns observed are significant but not conclusive proofs.

Open Questions

• Current knowledge suggests almost all numbers tend to shrink, but there is no definitive proof.
• The existence of a divergent trajectory or loops in positive numbers remains unproven.
• Negative numbers result in multiple independent loops, raising questions about their behavior.

Conclusion

• The Collatz conjecture serves as a reminder of the uncertainty and complexity in mathematics.
• It emphasizes the peculiar nature of numbers.
• The conjecture remains unsolved, highlighting the challenges in mathematical proofs.
• Educational Note: Learning through experimentation and problem-solving is crucial in math education.

Resources

• Brilliant.org: A platform for interactive learning in mathematics and STEM-related fields.