Re: [PrimeNumbers] prime is defined by a tautology
- On Wed, 14 Dec 2005, shuangtheman wrote:
> I was recently struck by the fact that the concept of primes is defined byFrom: Wojciech.Florek@...
> a tautology or circular reasoning. Prime is defined by calculation
> (divisible by itself and 1 only) that is itself defined by numbers or
Date: 12/14/05 18:12:10
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.
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.
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