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Collatz — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale.

    The question nobody can answer

    Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.

    part 1 · dynamics
  2. The share that provably comes down. The proportion of starting values that fall below their own start within k steps, plotted against k up to 12. The proportion rises towards one; at the largest k drawn it is 0.94.

    Almost every number comes down

    The Collatz conjecture is open and a great deal about it is not. Whether a number falls below its own start in the first few steps is decided entirely by its remainder on division by a power of two, and the share of numbers for which it happens can be counted exactly.

    part 2 · dynamics
  3. Downhill on average, and never on purpose. The logarithms of 4 Collatz orbits plotted against step number, each wandering upward and downward and each ending at one. A separate sample of four thousand starts gives an average fall of -0.15 per step.

    The heuristic that cannot be a proof

    There is a two-line argument that the Collatz conjecture is true, it is convincing, and everybody who works on the problem believes it. It also cannot be turned into a proof, and understanding exactly where it fails is more instructive than the argument itself.

    part 3 · dynamics
  4. The only fractions that could be a cycle's shape. A table of the convergents of the base-two logarithm of three, with the approximation error, the exact value of two to the n less three to the k, and that value as a fraction of three to the k.

    How short a cycle could be

    The drift argument cannot see cycles at all, which is why it is not a proof. What can see them is arithmetic — a cycle's shape has to be a fraction that approximates the logarithm of three to base two extraordinarily well, and there are very few such fractions.

    part 4 · dynamics
  5. Every parity pattern of length up to 12, and each occurring exactly once. A bar for each pattern length, showing the number of distinct parity patterns produced by all remainders of that power of two, which equals the number of remainders at every length.

    Every pattern happens exactly once

    Choose any sequence of odds and evens and there is exactly one residue class whose orbit follows it, and exactly one fraction that cycles through it forever. The Collatz conjecture is then the statement that only one of those infinitely many cycles is made of whole numbers.

    part 5 · dynamics

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