### More Mysteries of the Number 24

#### Posted by John Baez

For a long time I’ve been fascinated by the mysteries of the number 24: the way it shows up in string theory, the Leech lattice and Monstrous Moonshine, the 24-element binary tetrahedral group, the 24-cell:

and so on — even the fact that

$1^2 + 2^2 + \cdots \cdots + 23^2 + 24^2$

is a square number (a fact which turns out to be related to the Leech lattice). I’ve always dreamed of writing a book called *My Favorite Numbers*. In this book, chapter 24 would be longer than most.

Now I need your help!

Recently James Dolan and I were studying the Riemann–Hilbert correspondence — but that’s another story. In the process, he ran into something called “Kummer’s 24 hypergeometric functions”! I don’t know what these are… my only clue is a Wikipedia article which says:

The classical standard hypergeometric series is given by:

$\,_2F_1 (a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n} \, \frac {z^n} {n!}$

where $(a)_n = a(a+1) \cdots (a+n-1)$ is the rising factorial, or Pochhammer symbol. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence. This series is one of 24 closely related solutions, the Kummer solutions, of the hypergeometric differential equation.

So: does anyone know what these Kummer solutions are… and *why there are 24 of them*?

## Re: More Mysteries of the Number 24

Hi John,

Take a look at

Maier, Math Comp. 76 (2007) 811-843

http://www.ams.org/mcom/2007-76-258/S0025-5718-06-01939-9/S0025-5718-06-01939-9.pdf

this lists the solutions (see Table 1), the 24 is the order of the group D_3 that permutes these.