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Re: [PrimeNumbers] prime is defined by a tautology

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  • Kermit Rose
    ... From: Wojciech.Florek@amu.edu.pl Date: 12/14/05 18:12:10 To: shuangtheman Cc: primenumbers@yahoogroups.com Subject: Re: [PrimeNumbers] prime is defined by
    Message 1 of 4 , Dec 15, 2005
      On Wed, 14 Dec 2005, shuangtheman wrote:

      > I was recently struck by the fact that the concept of primes is defined by
      > a tautology or circular reasoning. Prime is defined by calculation
      > (divisible by itself and 1 only) that is itself defined by numbers or
      > primes.




      From: Wojciech.Florek@...
      Date: 12/14/05 18:12:10
      To: shuangtheman
      Cc: primenumbers@yahoogroups.com
      Subject: Re: [PrimeNumbers] prime is defined by a tautology


      10. But, at last, there are numbers which can be written in one and only
      one (taking into account commutativity) TRIVIAL way p=1.p.
      11. Conclusion 1. Number 1 is NOT a prime number!
      12. The next conclusion is well known (but so straightforward to prove):
      any natural (integer) number with non-trivial decoposition n=a.b
      (non-trivial means a,b>1; these number are composities, of course)
      can be written as a product primes in the UNIQUE way!

      Summing up this long list. I haven't used the term "prime" before defining
      "prime". The only distinguished numbers are 0 and 1. Neither of these
      numbers is prime! To define division and divisibilty we need numbers
      (mayby rational ones), but we do NOT need PRIMES. No tautology.

      Regards
      Wojtek




      From kermit@...


      Wojtek's comments suggest the following definition to me.

      Equivalent to the usual definition of prime is the definition:

      A positive integer > 1 is prime if it can be written as

      a difference of positive square integers in exactly one way.



      Kermit




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