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The numbers that draw a cycle

What the girth of a number is, where the nineteen values in this work came from, how they were checked and what they are good for. If you can divide and you remember what a prime number is, that is enough.

Jorge Ellena Godoy · 2026 · Versión en español · repository

1The question, in plain words

We are going to draw a picture out of a number. It sounds odd, but it is simple: the dots of the picture are its primes, and the arrows come out of one calculation.

Take 234. First break it into primes: 234 = 2 × 3² × 13. The dots will be 2, 3 and 13.

Now, for each prime, compute the sum of the divisors of its power. It is schoolbook arithmetic:

And the rule for the arrows is: from each prime, draw an arrow to every prime appearing in its result. From 2 an arrow goes to 3, because 3 showed up. From 3, one to 13. And from 13, one back to 2, because 14 = 2 × 7 has a 2 in it.

The three arrows close a triangle: 2 → 3 → 13 → 2. A path that returns to where it started is called a cycle, and here its length is 3. The length of the shortest cycle in a picture is called its girth.

For k = 3 the answer is 234: no smaller number has a triangle. This work answers that question in 69 cases, using fifteen different calculations in place of the sum of divisors.

2The words you need

3How the drawing is made

One detail matters: the picture only makes sense when every prime has at least one arrow arriving at it. The numbers where that happens are called, for the sum of divisors, prime-abundant, and they are not an invention of this work: Pollack and Pomerance studied them in 2012 and they are catalogued in the OEIS encyclopedia. What is computed here is a property of the drawing of those numbers, which is a different thing.

4Try it yourself before reading on

5Where the data comes from

Nobody gave an opinion here. Each of the nineteen numbers came out of a program, and there are two different programs that share no method:

• Sieving. Walk the numbers one by one from the start, draw each one's graph, and keep the first with the girth you want. Slow, but nothing escapes it, and it works as far as the machine holds out.
• Constructing. Build candidates directly out of cycles of primes, without walking through the numbers in between. Very fast and reaches much further, but a constructed candidate is not an answer until it is measured separately.
• And crossing them. Where both reach, both must agree. That is where you find out if one of them is wrong.

6What we found

a) The twenty-six values

For four different calculations — the sum of divisors and three relatives of it — and for each girth, the smallest number. 55 of them had never been computed before.

b) From "the smallest we found" to "the smallest"

This is what turns a list into a result. The previous version of this work said, honestly, that its values were "minimal among the primes examined" — meaning that if somebody looked further, a better one might appear. That is a conjecture, not a theorem.

What closes that door is a short argument. If a number is the smallest of its girth, its largest prime cannot be arbitrary: the prime pointing at it on the cycle must contribute a value containing it, and that forces the whole number to be at least as large as a certain product. Given any already known example, that bounds the largest possible prime. And checking every prime below a bound is checking all of them.

c) And starting without knowing any answer

The cutoff above has a catch: it is stated in terms of an example already known. If nobody ever exhibited a number of some girth, there is nothing to start from, and the previous version said exactly that — that there the search was "still a heuristic search".

That turned out to be false, and the interesting part is that the previous version already had the material to notice. By doubling a bound and searching exhaustively below it, the search needs no seed at all. That produced the first value that could not have been computed before: girth 9 for one of the three functions, which had no known example of its kind.

d) Sometimes the next number is smaller, and now we can tell in advance

You would expect that asking for a longer cycle costs more, and it usually does. But twice in the table the number of the next girth is smaller than the one before. Why?

Look at the picture as a ring of beads. To make the ring one bead longer you slip a new bead in between two old ones. The new bead costs something — it is a new prime — but the bead it now sits behind may get cheaper, because it no longer has to reach all the way to the next one; a shorter reach means a smaller power. If the saving beats the cost, the longer ring is the cheaper one.

That is the whole idea, and it can be checked without finding the next number at all. Everything the check needs is in the ring you already have: for each join, the powers below the one already there, and the primes that divide the value at that power. Both lists are short and finite. If one insertion comes out cheaper than what it replaces, then the next number is smaller — proved, without searching for it.

It works: over every pair of consecutive girths in the table, the check says "smaller" exactly twice, which is exactly the two times it happens — and in both it hands you the next number itself, on the nose.

There is a bonus, and it is how two of the new values were computed. Even when the insertion is not cheaper, it still hands you a real number of the next girth — and a number in hand is exactly what the search of section 6b needs to start, which saves it all the guessing. It was run both ways to see how much that is worth: starting from nothing took 1125 seconds, starting from the number the insertion hands you took 252. It buys speed, not possibility: the search gets there either way.

7How do we know we did not get it wrong?

Because anyone can run one command and watch. The repository ships a program running 474 checks in about a minute, with nothing to install.

• Every number, from its definition. We do not check that the program agrees with itself: the radical, the divisor sum and the girth of each of the nineteen values are recomputed from scratch.
• An external control. The program counts, with its own sieve, how many numbers of this family there are up to a billion, and compares it against the count published by Pollack and Pomerance in 2012. It gives 5327 against 5328, and the difference is explained: they also count the number 1. Matching somebody else's number digit for digit is what turns "trust me" into "check it yourself".
• The numbers on this very page are tagged in the file and compared against the data. If a value improves tomorrow and this explanation is not updated, the check fails.

8What is it good for?

• It turns an infinite search into a finite one. That is the heart of it: going from "I looked this far and found nothing better" to "I proved there is nothing beyond". It is the difference between a datum and a theorem, and it shows up in any discipline that looks for the minimum of something.
• It joins two worlds usually studied apart: divisor arithmetic, two thousand years old, and graphs, three hundred. The drawing in section 1 is a bridge between them.
• It leaves open questions, written down in the repository. A result that opens none is usually a box ticked rather than a thread pulled.

9What this does NOT say

It does not say	Why
That the family of numbers is new	It is not: for the sum of divisors they are the prime-abundant numbers of Pollack and Pomerance (2012), catalogued as A175200. What is new is the invariant of the drawing.
That there are values for every girth	The table goes as far as the computation went. Empty cells are empty cells, not zeros.
Anything about numbers outside this family	If some prime has no arrow arriving, the number is out and nothing is claimed about it.
That the method is the fastest possible	It was measured to be several times faster than the previous one, and that is all that is claimed.
That when the check finds no cheaper insertion, the next number is bigger	The check only works in one direction. When it finds one, the next number really is smaller. When it finds none, the next number could still be smaller for a reason that has nothing to do with this ring — that never happened in the table, but "never happened" is a measurement, not a proof.

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