Generator

The Collatz orbit of 27

A generator in the dynamics library, called 30 times across 5 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

collatz is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale.

`show: "stopping"`

`show: "parity"`

The first 5 steps, decided by the start. A table of the 32 residues modulo 32 and the parity string each produces over 5 steps of the Collatz rule. All 32 strings are distinct, and the ones that shrink the number are marked.

`show: "tree"`

The tree above one, 9 levels up. The reverse Collatz tree rooted at one, drawn to depth 9, holding 32 numbers. Each level doubles the numbers of the one below and adds the odd numbers whose tripling lands there.

`show: "bijection"`

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.

`show: "rationals"`

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

step 1 follows the rule ×178
every one of the 2 patterns of length 1 occurs exactly once ×12
the orbit of 27 reaches one ×7
a power of two is never a power of three ×1
a starting number is a whole number between 3 and 1000000 ×1
across a large sample of starts, half the steps are odd ones ×1
and 27 is the famous long one ×1
and by the largest run drawn it is above nine tenths ×1
and it is the one at one ×1
and no two residues give the same parity string ×1
and reaches a few thousandths within the range drawn ×1
and the average fall per step is the value the pairing predicts ×1
at least one pattern gives a whole number ×1
between two and four odd offsets, each at most eleven ×1
each convergent beats every fraction with a smaller denominator ×1
every number the reverse tree reaches does come back to one ×1
every orbit trends downwards in the logarithm ×1
every pattern of the length was solved ×1
most parity strings shrink the number by the end of the run ×1
no pattern with a negative denominator gives a positive whole solution ×1
one is already at one ×1
one length per start ×1
so the residues and the parity strings match one for one ×1
the continued fraction of log₂3 starts 1, 1, 1, 2 ×1
the denominator is always odd, so the solution stays where the map is defined ×1
the depth of the tree is a whole number between 3 and 12 ×1
the largest number of steps is a whole number between 4 and 15 ×1
the largest start is a whole number between 20 and 4000 ×1
the longest pattern length checked is a whole number between 6 and 14 ×1
the longest pattern searched is a whole number between 8 and 20 ×1
the map is a bijection at every length checked ×1
the multiplier condition predicts the drop, on a large member of the class ×1
the number of convergents listed is a whole number between 5 and 10 ×1
the number of steps is a whole number between 3 and 7 ×1
the ordinary rule has exactly one cycle at these lengths ×1
the ordinary rule is among those searched ×1
the parity of the first steps depends only on the start modulo two to the k ×1
the pattern length is a whole number between 4 and 10 ×1
the relative gap shrinks along the list ×1
the rule with minus one has at least three, which no drift argument distinguishes ×1
the share provably driven down grows with the number of steps allowed ×1
the starting number is a whole number between 1 and 100000 ×1
the tree is still growing at the depth drawn ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Dynamics

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.

Dynamics

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

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

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.

Dynamics

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.

The whole library · What the figures prove