# The miracle of 163

On page 227 of Symmetry and the Monster, Mark Ronan tells the story of Conway and Norton computing the number of independent _mini j-functions_ (McKay-Thompson series) arising from the Moonshine module. There are 194 distinct characters of the monster (btw. see the background picture for the first page of the character table as given in the Atlas), but some of them give the same series reducing the number of series to 171. But, these are not all linearly independent. Mark Ronan writes :

"Conway recalls that, 'As we went down into the 160s, I said let's guess what number we will reach.' They guessed it would be 163 - which has a very special property in number theory - and it was!

There is no explanation for this. We don't know whether it is merely a coincidence, or something more. The special property of 163 in number theory has intruiging consequences, among which is the fact that

$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925...$

is very close to being a whole number."

The corresponding footnote is a bit cryptic and doesn't explain this near miss integer.

"This special feature also yields a fact, first noticed by Euler, that the formula

$x^2-x+41$

gives prime numbers for all values of x between 1 and 40. The connection with 163 is that the solution to

$x^2-x+41=0$

involves the square root of -163."

So, what is really going on?

The _modular j-function_ has a power series expansion in $q=e^{2 \pi i \tau}$ starting off as

$j(\tau) = q^{-1} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + \cdots$

and classifies complex elliptic curves upto isomorphism, or equivalently, two-dimensional integral lattices upto a complex scaling factor. A source of two-dimensional integral lattices is given by the rings of integers $\mathbb{Z}.1 + \mathbb{Z}.\tau$ in quadratic imaginary extensions of the rational numbers $\mathbb{Q}(\sqrt{-D})$. So, perhaps one might expect special properties of the j-value $j(\tau)$ whenever this ring of integers has special properties.

Leopold Kronecker discovered in 1857 the remarkable fact that the modular j-function detects the class number of $\mathbb{Q}(\sqrt{-D})$. Recall that the class-number is a finite number measuring the amount by which the ring on integers $\mathbb{Z}.1 + \mathbb{Z}.\tau$ in $\mathbb{Q}(\sqrt{-D})$ fails to be a unique factorization domain. He proved :

The function-value $j(\tau)$ is an algebraic integer whose degree is the class number of the quadratic extension. In particular, if the ring of integers in $\mathbb{Q}(\sqrt{-D})$ satisfied unique factorization (or equivalently, is a principal ideal domain), then $j(\tau)$ is an integer!

Special instances of this theorem were already known. For example, the Gaussian integers $\mathbb{Z}.1+\mathbb{Z}.i$ satisfy unique factorization and Gauss knew already that $j(i)=12^3=1728$. He even knew that

$j(\frac{1}{2}(1+\sqrt{-163}))=(-640320)^3=-262537412640768000$

and because $\mathbb{Z}.1+\mathbb{Z}.\frac{1}{2}(1+\sqrt{-163})$ is the ring of integers in $\mathbb{Q}(\sqrt{-163})$ and as the absolute value of this j-value is near the value of $e^{\pi \sqrt{163}}$ we must be closing in on the solution of the riddle.

Charles Hermite noticed in 1859 this curious numerical consequence of Kronecker's theorem. For, if one takes $\tau = \frac{1}{2}(1+\sqrt{-163})$ and plugs this into the definition of $q=e^{2 \pi i}$ one gets the tiny number

$q = e^{\pi i - \pi \sqrt{163}} = - e^{-\pi \sqrt{163}}$ which is equal to -0.0000000000000000038089809370076523382...

So, all but the first two terms in the series expansion of $j(\tau)$ will be very small. For example

$196884 q = -0.00000000000074992740...$ and the next term $21493760 q^2$ is already 0.00000000000000000000000000031183868... and further terms will be even a lot smaller.

Combining this information with the Gauss-computed value of $j(\tau)$ we get that

$~(-640320)^3 = -e^{\pi \sqrt{163}} + 744 - \text{tiny number}$

whence the observed curious approaximation of

$e^{\pi \sqrt{163}} = 262537412640768000 + 744 -\text{tiny number} = 262537412640768743.99999999999925...$

What about other near misses which follow from Kronecker's result? Unfortunately there are only nine imaginary quadratic extension $\mathbb{Q}(\sqrt{-D})$ for which the corresponding ring of integers satisfies unique factorization, namely

$D=1,2,3,7,11,19,43,67,163$

and of course the near misses will be worse for smaller values of D. For example for the next two largest values one calculates

$e^{\pi \sqrt{67}} = 147197952743.99999866245422...$

$e^{\pi \sqrt{43}} = 884736743.99977746603490661...$

Reference :

John Stillwell, Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76

## Comments

Beautiful! Thanks for creating such a wondeful new blog.

What a wonderful surprise! Congratulations on your beautiful and carefullly done new blog!

Christine

1 - Could this near miss be due to 43, 67, 163 not being supersingular primes?

2 - Thanks for the pdf registry. I thought I had a computer dump of your blog, but it was only a print dump of 20 pages inclusive from 'bloomsday-end' to the fifth part of your NCG geometry course 'the NC manifold of a Riemann surface' without any diagrams or figures.

Kea & Christine : thanks! Though I love to restyle my blog from time to time it's a time consuming exercise so it is always nice if al least some people like the result.

Doug : as to 1. no, there are infinitely non-supersingular primes and just 9 p's such that the ring of integers in $$Q(\sqrt{-p})$$ is a principal ideal domain (which is the reason for the near miss). as to 2. the pdf-files are not perfect but will have to do for now. probably i will recycle/update some old posts when they are on-topic here (such as the Mathieu post and some on modular groups).

To all of you I apologize for having set my options so that anyone can post a comment provided they have one comment approved. So, only your first comment will not appear automatic. I learned the hard way it saves me time removing spam-comments.

The mathematical near miss is conceptually reminiscent of the 1999 Physics Nobel Award, from the Presentation Speech:

“... 't Hooft and Veltman tell us instead: calculate as if the number of dimensions were slightly less than four, four minus epsilon, i.e., 3.99999. This approach proved to be highly effective. The nasty infinities became less frightening. They could be collected, harnessed and interpreted ...”

http://nobelprize.org/nobel_prizes/physics/laureates/1999/presentation-s...

Notice these curiosities:

3 + 1 = 4 = 2^2

43 + 1 = 44 = (2^2) * 11

67 + 1 = 68 = (2^2) * 17

163 + 1 = 164 = (2^2) * 41

These are similar to but different from:

196884 - 1 = 196883 = 47 * 59 * 71

Yet it appears as though all of these numbers can be related to some form of supersingular primes.

I do not know if this is significant or merely happenstance.

Hi! Been away for a while, so I didn't see the redesign till today.

Love the new design! Just a small accesibility issue: in ffox 2.x for linux (cannot say for windows) compulsory fields in any form appear with light yellow background. So for the name, email and website (don't know exactly why also this one) I am forced to reading white text over light yellow background, which is not a very pleasant experience.

More on-topic: Very nice property of 163! I must have a really twisted mind, though, since first thing I thought about it was "Ohhh, I could use it for the arbitrary-precision computing practice for my CS students, and tell them that it *is* a whole number unless they are introducting carrying mistakes" and then thinking of the feeling of walking around the clasroom, smiling for myself, while they desperately try to find errors in their codes... Maybe I'll do it, at least it will show them not to believe something true just because the teacher told it :D

A108764 Products of exactly two supersingular primes (A002267).

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 77, 82, 85, 87, 91, 93, 94, 95, 115, 118, 119, 121, 123, 133, 141, 142, 143, 145, 155, 161, 169, 177, 187, 203, 205, 209, 213, 217, 221, 235, 247, 253, 287, 289, 295, 299

There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group....

I remember martin gardner wrote an april fool's joke in 1975 that announced a proof that e^{\pi \sqrt{163}} is an integer!

It is also the rank of the monstrous moonshine functions. This is not a coincidence, it is significant.

Gukov-Vafa: RCFT & complex multiplication arxiv:hep-th/0203213
Comm Math Phys 246 (2004) 181-210.
in which (page 21 line 9) is the remark that the class
number h(sqrt(-163)) = 1 implies the uniqueness of a certain
rational conformal field theory (RCFT) with chiral ring of dim 163.
Now for the monster group, M, and its moonshine functions MM, we
have (see Conway-Norton, Monstrous Moonshine Bull Lond.Math Society
1979, p 310 (1) and p319) the calculations giving the (column) rank
as 163 for the MM fns (conceived as vectors of their q-coefficients,
which are all rational integers). This 163 is obtained
from the 194 conjugacy classes of M from which we derive
172 classes of cyclic subgroups. There are 9 linear relations
between the corresponding 172 rational MM fns. Tables of these are
in McKay p 117-118 of CRM/AMS Proc. vol 30 ed. McKay & Sebbar 2001.
We also have a paper in Les Houches (2003) Proceedings II pp 373-386.
in which we note the possible significance in physics of the number
of classes in a finite group.
Irrational character table entries for M are the 22 2x2 cells of
complex quadratic entries. There are two classes, 27A & 27B that have
he same MM fn. but have centralizers of order 486 and 243. This
curiosity remains to be explained.