Mark Ronan's website
163, the Monster and
Number Theory
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The Monster, number theory and 163 The number 163
plays a special role in connection with the Monster
group. It — or rather the square root of -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 the
square root of ‑163 is given on page 227 in my book Symmetry and the Monster, where I point out that e to the power of |
Number theory To illustrate
the role of 163 in number theory, consider the equation y = x2 – x + 41. When x = 1,
2, 3, . . . , up to 40, it turns out that y is a prime number. Here are the first few.
The proof that y is a prime number uses the square root of ‑163,
which is involved in solving the equation x2 ‑ x + 41 = 0. The solutions a = The rational numbers extended
by the square root of -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. |
