# 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.