## Tuesday, August 18, 2009

### What is the least boring number (NOT the usual paradox)

What is the first boring natural number? I am NOT going to present that the first boring natural number is interesting crap, which may qualify as the most boring paradox.

The question is, of course, ill defined. I will define it a little better by only considering mathematical properties of numbers. (e.g., 7 is interesting because there are 7 days of the week will not work.) Here are my opinions, and my opinions of my opinions. I will write WEAK if I think the justification for calling that number interesting is weak. In those cases if you know a better one, then comment on it.
1. 1 is interesting as it is the multplicative identity.
2. 2 is interesting because it is the only even prime. Also the first prime.
3. 3 is interesting because it is the first odd prime. Also the first Mersenne prime.
4. 4 is interesting because it is the first non-trivial square. Also it is the first number that is the sum of two primes.
5. 5 is interesting because it is the first number that is the sum of two distinct squares and the first number that is the sum of two distinct primes. (WEAK)
6. 6 is the first perfect number (though there are so few perfect numbers that ALL of them are interesting.)
7. 7 is the first number such that the number of squares needed to add up to it is 4 (All numbers are the sum of 4 or less squares. There are an infinite number of numbers that require 4 squares: all of the numbers congruent to 7 mod 8.)
8. 8 is the first non-trivial cube.
9. 9 is the first non-trivial odd square. (weak)
10. 10 is the first number that is the sum of two distinct odd squares. First triangular number that is the sum of 3 squares. (weak)
have not been able to come up with anything interesting about 11. I could say that 11 is the first number that is the sum of 2 distinct numbers in 5 different ways. But that seems very weak: every number of the form 2n+1 is the first number that is the sum of 2 distinct numbers in n ways. Also, every number of the form 2n is the first number that is the sum of 2 numbers in n different ways. If we allowed that definition of interesting then all numbers would be interesting.

1. 5 is interesting also because Z[\sqrt{-5}] isn't a UFD; 5 is the smallest integer with this property.

A_5 is the smallest nonabelian simple group.

In group theory, 11 is related to the smallest Mathieu group (the smallest sporadic simple group) -- it's one of only four special cases of 4-transitive permutation groups.

8 is extremely interesting (more or less because it divides 24, which is probably the MOST interesting number) -- the E_8 lattice is one example.

My vote for the first boring number is 9, which doesn't have nearly as many interesting algebraic, geometric, etc. properties as the numbers around it.

2. How many times do these numbers appear in Sloane's Encyclopedia of Integer Sequences?

I searched for 1, 2, ..., 31. As you might expect there's a decreasing trend. But 9 is the first number to appear less often than its successor (9 appears 53266 times, 10 appears 57972 times). This doesn't necessarily mean 9 is uninteresting, though; it might just mean 10 is interesting.

And of course there are flaws in this method. For example there are 10 hits for 196884 -- which occurs in the study of the Monster group -- but most of them are really "the same" as there are a lot of trivial variations on the same sequence, or sequences that don't include 196884 at all but include it in the explanatory text.

3. Yes, that's an unusual paradox: "The least boring number is boring"

4. In finite geometry, 9 is the order of the smallest Non-Desarguesian projective plane, and also the first order for which the projective plane is non-unique.

5. The integers -1,0,1 are extremely interesting because of their role in fields. The integer 2 is extremely interesting, especially in computer science, because it is the least prime and because (-1)^2 = 1. No other integers are anywhere near as interesting as {-1,0,1,2}, so I claim 3 is the least boring natural number.

By the way it's silly to say that 2 is interesting because it is the only even prime. That is equivalent to saying that 2 is the only prime divisible by 2, which would be true if "2" were replaced by any other prime! The number 2 is so important that there are special words for numbers that are divisible by it, but that is a consequence of its being interesting, not the reason.

6. 7. 4 is the first composite number

8. There's also this list...

9. It's worth mentioning that numbers that require four squares aren't just those congruent to 7 mod 8, but all those of the form 4^a (8b + 7).

I happen to collect lists of uninteresting numbers on my website, so I found this post particularly interesting.