Lecture Notes: The Collatz Conjecture

Introduction

• The Collatz Conjecture, also known as the 3x+1 problem, is considered one of the most dangerous problems in mathematics.
• Mathematicians warn against spending time on it due to its infamous unsolved status.
• Paul Erdos remarked on the immaturity of mathematics to handle such questions.

How It Works

• Starting Point: Pick any positive integer.
• Rules:
  • If the number is odd, multiply by 3 and add 1.
  • If the number is even, divide by 2.
• Example: Starting with 7, the sequence eventually enters a repeating loop of 4, 2, 1.
• Conjecture: Every positive integer will eventually end up in the 4, 2, 1 loop.

Names and History

• Known by various names including Ulam conjecture, Kakutani's problem, Thwaites conjecture, Hasse's algorithm, Syracuse problem.
• Originates from 1930s, attributed to German mathematician Luther Collatz.

Mathematical Insight

• Hailstone Numbers: Numbers exhibiting up-and-down behavior, like hailstones.
• Stopping Time: Number of steps to reach 1 in the sequence.
• Variant Paths: Different numbers have vastly different stopping times.

Challenges and Mathematicians' Approach

• Often considered a distraction or a puzzle with no progress.
• Jeffrey Lagarias: Leading authority advises against academic focus.
• Terry Tao: Found that "almost all" numbers have sequence with an arbitrarily small term.

Statistical Analysis

• Benford's Law: Distribution of leading digits in sequences follows this statistical rule.
• Growth Misconception: Although sequences seem to grow, they actually trend down because odd numbers become even, reducing size over steps.

Visualization and Graphs

• Directed Graphs: Numbers form a tree-like structure connecting back to 4, 2, 1.
• Coral Representation: Altered visualizations show organic patterns.

Possibility of Counterexamples

• Could be numbers forming a loop or going to infinity not connecting to the main graph.
• No such examples have been found despite extensive computational checks.

Attempted Proofs and Advances

• Riho Terras (1976): Showed sequences dip below their starting value often.
• Terry Tao's Progress (2019): Almost all sequences go below any function that goes to infinity, though not a complete proof.

Philosophical and Mathematical Implications

• Even with brute force checking up to 2^68, no conclusive proof.
• Possible parallel to concepts in the halting problem (undecidable problems).
• Mathematical exploration often reveals complexity in seemingly simple problems.

Conclusion

• The problem remains unsolved, showcasing both the beauty and challenge of mathematics.
• Highlights the complexity of number theory and chaotic behavior in mathematics.
• Ends with a discussion of learning resources and ongoing inquiry.

Additional Resources

• "Brilliant" platform mentioned as a sponsor, offering interactive learning experiences for deeper understanding.