163 and the Monster
The number 163 plays a special role in connection with the Monster group. It, or rather \(\sqrt{-163}\), also plays a special role in number theory. It is not yet known if there is a direct connection between these two appearances, though other connections between the Monster and number theory have been proved and explained under the general heading of Moonshine.
The Monster
The significance of 163 for the Monster has to do with its character table, which has 194 columns. Each column yields a ‘moonshine function’, but these funtions are not all linearly independent, and the space they span has dimension 163.
Moonshine
The Moonshine phenomenon connects the Monster with the \(j\)-function in number theory, and a connection between this and \(\sqrt{-163}\) is given on page 227 in my book Symmetry and the Monster, where I point out that \(e\) to the power of \(\pi\sqrt{163}\) is extremely close to being a whole number.
Number theory
To illustrate the role of 163 in number theory, consider the equation \(y = x^2 – x + 41\). When \(x = 1, 2, 3, \dots\) , up to 40, it turns out that \(y\) is a prime number. Here are the first few.
| x | y |
|---|---|
| 1 | 41 |
| 2 | 43 |
| 3 | 47 |
| 4 | 53 |
| 5 | 61 |
| 6 | 71 |
| 7 | 83 |
| 8 | 97 |
The proof that \(y\) is a prime number uses \(\sqrt{-163}\), which is involved in solving the equation \(x^2 – x + 41 = 0\). The solutions \(\alpha = (1 + \sqrt{-163})/2\) and \(\beta = (1 – \sqrt{-163})/2\) (obtained by using the quadratic formula) can be used to write \(y\) as the product of two factors \((x-\alpha)(x-\beta)\). When \(x\) is an ordinary integer, the factors of \(y\) are algebraic integers that lie in the field of rational numbers extended by \(\sqrt{-163}\). The algebraic integers in this field admit unique factorisation into primes, and this fact can be used to prove that \(y\) is a prime number when \(x\) is an integer from 1 to 40.
The rational numbers extended by \(\sqrt{-n}\) do not normally admit unique factorisation; it only happens when \(n\) is one of the following: 1, 2, 3, 7, 11, 19, 43, 67, and 163, which makes 163 rather special.