Overview
This lecture explores the Collatz conjecture—an unsolved mathematical problem involving a simple sequence with unpredictable behavior, its history, analyses, and the challenge of proving or disproving it.
The Collatz Conjecture: The Problem
- Start with any positive integer; if odd, multiply by three and add one; if even, divide by two.
- Repeat the process; the conjecture claims all numbers eventually reach the loop 4 → 2 → 1.
- Also called the 3x+1 problem, Ulam conjecture, Kakutani's problem, Thwaites conjecture, Hasse's algorithm, or Syracuse problem.
- Despite simplicity, the problem has resisted proof and is considered risky for mathematicians to tackle.
Behavior and Patterns in Sequences
- Sequences from different starting numbers show unpredictable, "hailstone"-like paths before settling into 4 → 2 → 1.
- Some numbers, like 27, grow very large before falling to one, with stopping time varying widely.
- The process appears random, resembling geometric Brownian motion; statistical analysis shows an overall downward trend.
Statistical and Digit Analysis
- Looking at leading digits of sequence numbers produces a distribution matching Benford's law, seen in various real-world data sets.
- Benford's law cannot determine if all numbers eventually reach one.
Mathematical Investigations and Results
- Multiplying an odd number by three and adding one always yields an even number, which after repeated division by two, typically shrinks.
- On average, moving from one odd number to the next in a sequence multiplies the value by 3/4, favoring shrinking over growth.
- Directed graphs model sequence paths; if conjecture holds, all numbers connect to the 4,2,1 cycle.
- No divergent paths or alternate loops have been found up to 2^68 (~300 quintillion numbers).
Approaches to Proof and Limitations
- Proving every sequence eventually decreases in value would prove the conjecture, but this hasn't been achieved.
- Riho Terras and others have shown "almost all" numbers get below initial value, but "almost all" does not mean "all."
- Terry Tao proved that almost all sequences contain arbitrarily small numbers, getting very close to a proof but not closing the case.
- Analogies with other disproven conjectures (e.g., Polya’s) highlight the limitations of brute-force checking.
Possibility of Counterexamples or Undecidability
- It remains possible that a counterexample exists far beyond current computational reach, or multiple independent loops exist (as proven for negative numbers).
- The problem could theoretically be undecidable, as related systems are Turing-complete and subject to the halting problem.
Philosophical Perspective
- The Collatz conjecture illustrates the unpredictable complexity of numbers and the limitations of mathematical knowledge.
Key Terms & Definitions
- Collatz conjecture (3x+1 problem) — The claim that every positive integer sequence under specific rules eventually reaches 1.
- Hailstone numbers — Numbers in a Collatz sequence, named for their up-and-down behavior before settling.
- Stopping time — Number of steps it takes a sequence to reach 1.
- Benford's law — A logarithmic distribution of leading digits found in many data sets.
- Geometric Brownian motion — A process resembling random movement with a trend, used as an analogy for Collatz sequences.
- Directed graph — A diagram representing number transitions in a Collatz sequence.
- Turing-complete — Capable of performing any computation, implying potential undecidability of problems.
- Halting problem — The question of whether a computation will eventually stop.
Action Items / Next Steps
- Review definitions and process for the Collatz conjecture.
- Try generating and analyzing Collatz sequences for different starting numbers.
- Reflect on the role of statistical evidence versus proof in mathematics.