Interesting numbers in Maths [closed]

Question

Mathematics is beautiful. Many numbers are interesting such as

1729 = 1³ + 12³ = 9³ + 10³

Please list out the interesting numbers and reasons to make number interesting

Answer 1

All positive integers are interesting. Suppose they weren't. Let k be the lowest non-interesting positive integer. That is an interesting property. Therefore contradiction. Generalisation is left as an exercise to the reader.

Answer 2

$163$ is interesting because it's the largest Heegner number. This means that it's the largest positive integer $d$ such that the ring of integers in $\mathbb{Q}(\sqrt{-d})$ has unique factorization. If you don't know what that means, two other elementary consequences of $163$ being a Heegner number are that

$n^2 + n + 41$

is prime for $n = 0, 1, 2, ... 39$ and that

$e^{\pi \sqrt{163}}$

is extremely close to an integer.

Answer 3

I always liked 23 (or more precisely, -23) because Q adjoin the square-root of -23 is the first quadratic imaginary field whose class number is not a power of 2; in fact, it's class group is of order 3.

Answer 4

$196883$ is interesting because it's the smallest dimension of a faithful complex representation of the Monster group. Roughly speaking, a group is a collection of symmetries of something. A simple group is a group which can't be decomposed further into smaller groups; to borrow a phrase, it's an "atom of symmetry." There is a classification of these atoms with several infinite families and 26 exceptions, and the Monster group is the largest of these exceptions. And the simplest way to write it down is using $196883$-by-$196883$ matrices.

$196883$ was crucial to McKay's discovery of Monstrous moonshine, which began with the observation that the j-invariant, an important function in number theory, has a Taylor series in which $196884$ appears. The story of this discovery is explained in Mark Ronan's wonderful book Symmetry and the Monster, which I highly recommend.