# Interesting numbers in Maths [closed]

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

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## closed as not a real question by Jamie Banks, Akhil Mathew, Larry Wang, BlueRaja - Danny Pflughoeft, Ben AlpertJul 30 '10 at 2:30

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numbergossip.com –  Qiaochu Yuan Jul 28 '10 at 7:10
Great link Yuan –  pramodc84 Jul 28 '10 at 7:12
1729 is also a Carmichael number. –  starblue Jul 28 '10 at 7:25
Start another thread called un-Interesting threads on Mathunderflow... –  user126 Jul 28 '10 at 7:59

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.

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All rational numbers are interesting, for the same line of reasoning. To say that all real numbers are interesting you need the Axiom of Choice, unfortunately. –  mau Jul 28 '10 at 7:33
"Calculus III is left as an exercise for the reader." –  Vortico Jul 28 '10 at 16:57
@mau: only if there's something interesting about your particular choice of well-ordering for the reals. –  Simon Nickerson Jul 29 '10 at 22:07
Maybe it would be fun to suggest the smallest positive uninteresting real. –  marshall Dec 8 '13 at 16:51

$163$ 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.

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$196883$ 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.

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There is a faithful representation of dimension one less (196882) over F_2, and matrices for this representation have actually been computed. –  Simon Nickerson Jul 29 '10 at 22:03

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.

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