Quote:
Originally Posted by Jeff Root
It is clear to me that the number 'one' is a prime number (since it is not the product of other integers) ...
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a) Is 0.839 the product of other integers?
No.
So it's prime?
No, because by "other" you imply that the number in question is an integer.
b) So is 5 a product of
other integers?
Yes. (1 and 5, and 1 isn't 5, so it's an "other" integer.)
So it's not prime?
No, because you of course expect us to already know the "1
and itself" part of the
usual defintion, so "other" does not exclude 1 or itself.
c) So is -5 a product of
other integers?
No (using b) ). (Just 1 and itself.)
So is -5 a prime?
No (not usually
*); you possibly meant "natural numbers" not "integers".
Now, I'm not meaning to critique your short-hand definition. I know of course you didn't intend that to be a complete or fully accurate description, and I'm not playing semantic games (though possibly showing that semantic quibbling is counter-productive).
My point here is simply that something in maths needs to be consistently understood so that all mathemeticians and ordinary folk like me can all understand the same thing from them.
Quote:
Originally Posted by Jeff Root
..., but for convenience in certain cases in number theory, an ad hoc addition was made to the definition to specify that it is 'not a prime'. By nature, it is prime. By definition, it is not prime.
-- Jeff, in Minneapolis
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Primes are a kind of number. The definition of the set of those numbers must accurately describe that set. The set does not exist
because of the definition. We counted 1, 2, 3, ... long before we called them the set of "natural numbers". Currently (yes, opinion has changed) 1 is not a prime, thus the definition must exclude them.
How is 1 a prime by "nature"? By what standard is that judged? (The contention of the OP of this thread was that a definition that is "simpler" is "preferred". My counter the whole time has been that a "simpler" definition is not preferred if it is incorrect; and that besides - exclusion of 1 as a prime made more subsequent defintions "simpler" than it made more "complex", so even by the claim in the OP it is better that 1 not normally be considered prime.)
You say "for convenience" and "ad hoc addition". I would say "correction", "refinement" and "improvement".
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A note on negative primes:
http://primes.utm.edu/notes/faq/negative_primes.html