Quote:
Originally Posted by aastrotech
In the wiki fundamental theorem of aritmatic "There are natural extensions of the hypothesis of this theorem, which allow any non-zero integer to be expressed as the product of "prime numbers" and "invertibles". For example, 1 and -1 are allowed to be factors of such representations (although they are not considered to be prime)."
Which is not only a contradiction in terms but also a bit of circular reasoning.
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(My underline.)
(Full Wiki text:
http://en.wikipedia.org/wiki/Fundame..._of_arithmetic)
An extension is not a contradiction, nor is it circular. It's no different than sometimes allowing 1 to be used as a prime number (which I don't think anyone has disputed can be done).
Are you now wanting to re-write the fundamental theorem of arithmetic
as well as the standard definition of primes?
To force 1 to be normally considered a prime number, you'd also now have to force this '1 and -1 allowed as factors' extension of the fundamental theorem of arithmetic to be "normal". (And that now contradicts your original "simpler=preferred" contention.)
Quote:
Originally Posted by aastrotech
How about "Suspect Primes" to remind the user that any theory or assumption derived from them is suspect?
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Oh my goodness! If it were so clear that exclusion of 1 from the set of primes would make any theorem that uses that set "suspect", why on Earth do you think 1 would have been excluded in the first place? Do you think mathemeticians just decided not to "like" 1 and threw it out of the set?
Quote:
Originally Posted by aastrotech
The only thing that I see from that thread as useful to this thread is to point out that the post that triggered this exposition; "one is not a prime", might have been more accurate had it said "one is not a conventional prime".
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It should be quite clear by now that the standard defintion of prime number in current use excludes 1. Thus it was a fair comment. Instead of trying to argue that 1
is a prime and the standard definition is wrong, you could simply have noted that you prefer to use the older defintion.
Post #92 sums it up very well.