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[Computational Complexity] What is the least boring number (NOT the usual par...

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  • GASARCH
    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
    Message 1 of 1 , Aug 18, 2009
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      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.

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      Posted By GASARCH to Computational Complexity at 8/18/2009 09:05:00 AM
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