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.

10 comments:

  1. http://www2.stetson.edu/~efriedma/numbers.html

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  2. 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.

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  3. 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.

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

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  5. 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.

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  6. 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.

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  7. 4 is the first composite number

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  8. 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.

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