Concept

Collatz conjecture

The claim that halving every even number and tripling every odd one and adding one always leads to one. Every number anybody has tried does, no proof exists, and the two ways it could fail — a cycle or an escape to infinity — need completely different arguments.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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

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.

dynamics · Collatz

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.

dynamics · Collatz

How high an orbit climbs

The Collatz orbit of 27 climbs to 9,232 before it falls, and below ten million the champion, 6,631,675, climbs to sixty trillion — almost exactly the square of where it started. That is not a coincidence of the two numbers but a law of the whole family: an orbit climbs to x times its start with a chance of about one in x, and the exponent is one because half of three-halves plus half of a half is exactly one. Among N starts the luckiest climbs about N-fold, and so the records grow like the square of the start.

dynamics · Collatz

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive search Periodic orbit Bijection Collatz Continued fractions Diophantine approximation Drift Expectation Heuristic Invariant Modular arithmetic Random walk

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