On Primes, Kissing Numbers & the Monstrous Moonshine
What the densest sphere packings know about prime numbers, and how far that leads toward the Monster
Key message: The underlying paper to this article proves that the minimal generation lengths of prime ideals in the densest sphere packings E8 and the Leech lattice are strictly determined by number-theoretic residue classes, showing that 91% of the Leech lattice’s kissing shell consists of prime generators directly linked to the algebraic structure of the Monster group.
Put a ball on a table and ask how many balls of the same size can touch it at once. In the plane the answer is six, the familiar honeycomb. In three dimensions it is twelve, a question Newton argued about. In eight dimensions the answer is 240, and in twenty-four dimensions it is 196 560.
These two are special. In both dimensions there is a single, perfectly regular arrangement that achieves this kissing number and packs space as densely as possible: the lattice E8 and the Leech lattice. Both records are now proved theorems.
These two lattices are among the most symmetric objects in mathematics. The symmetry group of the Leech lattice leads, by a short chain of constructions, to the Monster, the largest of the sporadic simple groups. And the Monster, through “monstrous moonshine”, is tied to the coefficients of one of the oldest functions in number theory, the modular j-function.
So a natural question is whether prime numbers have a place in all of this. Not in a vague sense. Can you point at a prime and say where it sits in the Leech lattice?
That is the question of my new paper, First Shells of Prime Ideals in Cyclotomic Lattices. This article is about what the answer means, not about the proofs. The paper and all code are open (link at the end).
A lattice that is also a number system
The starting observation is old, and it is the key to everything else. E8 and the Leech lattice are not only arrangements of points. They can be written as number systems.
Take the complex numbers built from a root of unity: for E8 a 30th root, for the Leech lattice a 35th root. Their integer combinations form a ring of “cyclotomic integers”. You can add and multiply in it as with ordinary integers. With the right notion of length, the points of this ring are exactly the points of E8 or of the Leech lattice. The 240 or 196 560 kissing points then become particular numbers in that ring.
In such a ring, primes do not disappear. They split. An ordinary prime p breaks into “prime ideals”, and each prime ideal is generated by numbers, which are points of the lattice. So every prime gets a set of points that represent it, and among them there is a closest one to the centre.
I call the distance of that closest point the first shell of the prime. Rescaled by the size of the prime, it becomes a single number, ρ, which tells you how efficiently a prime manages to sit close to the centre. ρ = 1 means the prime sits as close as the geometry of the lattice allows.
E8: the remainder decides
For E8 the picture is complete, and it is surprisingly clean. Every prime larger than 5 leaves one of eight remainders when divided by 30. These are the same eight numbers that label the corners of the 16-cell in my earlier books. The remainder alone decides how the prime behaves:
- For remainders 7, 13, 17 and 23 the prime is rigid: it sits at exactly the closest possible distance, and the set of its points is a perfect scaled copy of E8 itself.
- For 11 and 19 the first shell follows a golden law. It is read off from the way p can be written as a² + ab − b², the number system of the golden ratio. Fibonacci appears inside E8 without being put there.
- For 29 it is a sum of four squares inside a smaller lattice shaped by the 16-cell.
- For remainder 1 there is no simple formula. The values scatter in a pattern governed by Hecke’s equidistribution theorem from 1918.
And in every class ρ ≤ 1. A prime never sits farther out than the densest arrangement allows. The paper proves this in three independent ways. One proof can be checked by hand, one uses certified computation, and one rests on Blichfeldt’s theorem from 1935 that E8 is the densest lattice in eight dimensions.
The Leech lattice: primes on the kissing shell
In twenty-four dimensions the same idea gives a first striking fact. Of the 196 560 kissing points, 178 920 represent primes or products of primes. So primes such as 71 and 211 do not just live somewhere in the Leech lattice. They lie on the kissing shell itself, among the balls that touch the central one.
For most remainders modulo 35 the paper proves exact laws for the first shell. Some classes are rigid again. Some follow the same golden law as E8. For others the answer is the corner of an explicit polytope, found by a method I call the facet criterion. The method turns an infinite search into finitely many exact checks. Its hardest case needed 194 different facet types.
Two classes remain open: the primes with remainder 1 or 34 modulo 35. There, the space that has to be searched has eleven dimensions, and none of the methods reaches that far yet.
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Interpretation: three layers
The most useful thing to come out of the work is not a single theorem but a way of seeing why the theorems hold. There are three layers.
Geometry. E8 and the Leech lattice are the densest lattices in their dimensions. That alone forces every subset of them shaped like a prime to be no denser than the whole. This part comes for free.
Arithmetic. The real question is whether the shortest point inside a prime’s region actually represents the prime, or is merely a point that happens to lie there. In E8 the answer is always yes, and the reason fits in one line: there are no small numbers in the ring that could get in the way. The smallest non-trivial “number size” is 16, which is too large to spoil anything. In the Leech lattice the smallest one is 71, and that is not large enough for the same argument.
Symmetry. The lattice has symmetries that mix the different prime ideals of one prime. These make the first shell depend only on the prime, not on which of its ideals you look at, and they reduce the work in the hard cases.
The arithmetic layer is where the open problem now sits. It can be stated without any polytopes or tori:
Conjecture. In the Leech lattice, the shortest point of every prime ideal represents that prime ideal.
I checked it for 1674 prime ideals, without a single exception. If it is true, the open classes close at once, because the Cohn–Kumar theorem (2009), which says that the Leech lattice is the densest lattice in 24 dimensions, then does the rest.
There is a telling contrast. For lattices that are not the densest in their dimension, such as the family A_{p−1}, the bound ρ ≤ 1 fails, infinitely often, for every prime p from 7 to 43. In some of them the shortest point of a prime ideal is indeed not a representative. Primes behave best exactly where the packing is best.
And the Monster?
This brings me to the question I was asked most often while writing: so have we connected the primes to monstrous moonshine?
The honest answer is: partly, and it is important to say which part.
Here is what is true and proved. Multiplying by the 35th root of unity is itself a symmetry of the Leech lattice. Inside the symmetry group of the Leech lattice, Conway’s group Co₀, it has a fingerprint called its Frame shape, 1·35/(5·7). Through the correspondence of Conway and Norton, this fingerprint belongs to a specific element class of the Monster, called 35B. Its “moonshine function” is a clean expression in Dedekind’s eta function, η(5τ)η(7τ)/(η(τ)η(35τ)), minus 1.
The new part is this. The symmetries that the paper had to construct to handle the primes are twisted versions of the Galois maps of the number system. They turn out to be, up to roots of unity, all of the symmetries that respect the 35th root of unity: 1680 of them. So the remainder of a prime modulo 35 picks out a definite family of Leech symmetries. Each family carries Frame shapes, and these point to Monster classes. In the rigid classes all 560 symmetries share one fingerprint.
What this gives is a dictionary. Every prime can be translated, through its remainder, into a set of Frame shapes of Conway’s group, and for most remainders also into a set of Monster classes; every step of the translation is a theorem. The translation does not run backwards to individual primes: a class that occurs determines an infinite family of primes, defined by remainders modulo 35, and most Monster classes do not occur at all.
What it does not give, yet, is a bridge. A bridge would mean that the numbers we computed for an individual prime, its first shell, show up in the moonshine functions themselves, or are predicted by them. I have not found such a relation, and I have not seriously tested for one. Until someone does, it would be wrong to say that the primes have been connected to the Monster in the way that moonshine connects the Monster to the j-function.
It is also fair to say what is not new. That the Leech lattice can be written over the 35th roots of unity goes back to Craig in 1978. The correspondence between that symmetry and the Monster class 35B is in the tables of Conway–Norton (1979) and Lang (1989). What the paper adds is the link from the prime-by-prime laws to that symmetry.
A coincidence worth defusing
One number tempts every reader: 71. It is the smallest prime that splits in the Leech number system, and it is also a factor of 196 883, the dimension of the smallest non-trivial representation of the Monster (196 883 = 47 · 59 · 71). Is that a sign?
№71 is simply the smallest prime that leaves remainder 1 on division by 35, since 71 = 2 · 35 + 1. That is why it is the smallest size of a prime ideal in this ring, and it has nothing to do with the Monster. I mention it because this is exactly the kind of coincidence that can carry a theory away. In the paper it is listed as a coincidence.
How the work was done
Every statement in the paper carries a label: proved, proved relative to a cited theorem, computer-assisted, numerical, conditional, framework-internal, open, refuted or withdrawn. There is a separate section of negative results and corrections. It includes ideas that did not work, a claim I had to withdraw, and errors in my own test scripts that were caught and fixed along the way. One example: a naive expectation that larger open search spaces come with more Frame shapes turned out to be false. All computations are in an open archive with expected outputs and checksums, so anyone can re-run them.
None of this makes a physical claim. Moonshine is sometimes discussed in connection with string theory. This paper is pure mathematics, and the dictionary above is a statement about lattices and groups, nothing more.
What it implies
For number theory, the work turns a geometric question into an arithmetic one. Whether a prime sits as close to the centre as possible depends on whether the shortest points of its ideals represent it. That question makes sense for every number system of this kind, and the computations show that the answer really varies: yes for E8, D4 and, so far, the Leech lattice, and no for some others.
For the larger programme behind my books, it is a confirmation of a principle I keep testing: a geometric picture of the primes is only worth something if it computes things that can be checked independently. Here the 16-cell remainders modulo 30 decide the laws in E8, the golden ratio appears on its own, and the densest lattices turn out to be the most “prime-friendly” ones.
For the Monster, the door is open but not walked through. There is a precise dictionary from primes to moonshine classes. Whether anything flows back across it, whether a moonshine function knows the first shell of a prime, is now a well-posed question. I would be glad to hear from anyone who wants to try.
The paper, its LaTeX source and the complete verification code are available on Zenodo: https://zenodo.org/records/23293398
