# 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.