Mathematical Midgets: Fave Small Numbers & Why

[Deveno]

What are your favorite small numbers? Why?

Anyone can post in this thread, but the rules are:

1. The numbers must be single digit (you can cheat if you use hexadecimal, hint, hint).
2. 1 doesn't count (it's not prime, so go away).
3. If a closed form is not available, you can only use 20 symbols or less to describe it (words will count as one symbol).
4. Extra points if the number is natural.

A famous story is told about some Indian guy riding in a taxi-cab to meet a "real" mathematician (the famous Mr. Hardy. Bow down before your English masters!) who thought the taxi-cab number was interesting, after all. He was wrong, and *that* number is MUCH TOO BIG.

At the moment, my favorite number is 3...it's prime, it's very odd, and I have yet to unravel it's deepest mysteries (why does period 3 imply chaos? I really would like to know...). The fact that $\Bbb Z_3$ can be written as {-1,0,1} saves me lots of time while typing, because when I need some larger digit, I often have to look up what it is in a numerical dictionary (yes...I am *that* lazy).

3 is also my favorite counter-example...in group theory I often use $S_3$ to disprove mistaken "theorems" (such as the infamous Converse Lagrange Theorem), and my personal Anti-Riemann Hypothesis is: the smallest exception to the Riemann Hypothesis is of the form:

$\frac{1}{2} + 3ki$

for some real number $k$).(P.S. Don't take what I say too seriously. I lie. A LOT.).

 

[mathbalarka]

5.

Element of the first twin prime pair, conjecturally the smallest and only untouchable odd number, smallest $n$ such that $S_n$ has no solvable tower of subgroups and as well being the highest degree of my favorite polynomial (quintic). The later property can also be stated as : the highest degree of the general polynomial that is resolvable in terms of elliptic functions and order 1 thetas. (As the degrees increase, more elliptic as well as additional hyperelliptic and higher thetas are needed).

The smallest (or any) exception to RH must have it's imaginary part not $\frac12$, just as a note.

Good thread, by the way!

 

[Opalg]

What are your favorite small numbers? Why?

This reminds me of a famous lecture that J-P Serre gave at Harvard University in 2007 on the numbers 2 to 8. The only part that I remember hearing about at the time was the section about 6, which was devoted to a discussion of the fact that $S_6$ is the only symmetric group with outer automorphisms.

You can find a report on the lecture here.