Very nice reply. You didn't tear down my argument, but I think you
showed it to be weak. I can't tear down your argument, but I think
I can show it to be weak.
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Originally Posted by pzkpfw
Quote:
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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. |
I was alluding to my post #82 in which I gave my definition:
Quote:
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Originally Posted by Jeff Root
A prime number is a positive integer which is not the product of
other integers.
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I'm actually sufficiently ignorant of the subject not to know how
factoring of negative numbers is handled. If you can believe such
ignorance is possible. But it appears to be safely ignorable as long
as we stick strictly to the question of whether the number 'one' is
prime or not. So I recognize that it appears to be necessary to
restrict primes to positive numbers, but can't say exactly why.
My ignorance about this is obviously germaine to my argument, but
I hope it doesn't get in the way of my argument.
It seems obvious that my definition of prime numbers captures the
intended idea of what prime numbers are. If modern mathematics
chooses to tweak that definition, for whatever reason, it moves the
definition away from the intent. By eliminating 'one' from the set of
prime numbers, the modern definition deviates from the fundamental
concept of what a prime number is.
Quote:
Originally Posted by pzkpfw
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.
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My definition bypassed the languge about "1 and itself", by design.
I consider that language superfluous. I think it was put there as a
clarification, rather than as an essential part of the definition.
When I say "other integers" I mean precisely "other integers".
Five is not a product of other integers.
Quote:
Originally Posted by pzkpfw
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".
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If the discussion can be limited to natural numbers (positive integers,
as I said in my definition), then I will be much more comfortable in my
argument. Thank you! (This relates to my question earlier in the
thread as to whether prime numbers or negative numbers were the
first to be invented/discovered, and is suggested by a comment in
the link you provided.) I meant "integers", trying to be as inclusive
as possible. I think that "integers" works. Five is not the product
of any other integers, for example, including negative integers. But
you were responding to my shorthand, as you say, not my definition
which limited primes to positive integers.
Quote:
Originally Posted by pzkpfw
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.
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No argument.
Quote:
Originally Posted by pzkpfw
Quote:
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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.
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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. |
This really sounds like you are making my argument for me.
Quote:
Originally Posted by pzkpfw
How is 1 a prime by "nature"? By what standard is that judged?
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I think that my definition, that a prime number is a positive integer
which is not the product of other integers, captures the intended
meaning of "prime number". This definition, primitive and not the
currently universally-used definition, includes 'one' as a prime.
Quote:
Originally Posted by pzkpfw
(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.)
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I suspect that the orginal poster used the idea of simplicity as the
closest idea to what he meant. I used the idea of naturalness.
Both are very subjective. Okay-- mine is horribly subjective.
I would not interpret his preference for simplicity too literally.
Which definition is "correct" in this case just means which definition
has been accepted by convention. So being correct doesn't mean
that it is the best definition or the one that people naturally came
up with when they started thinking about relationships between
different numbers.
If you are right that the convention of excluding '1' as prime made
more subsequent defintions "simpler" than it made more "complex",
then that shows the convention to have some utility. Perhaps
enough utility to justify it.
Quote:
Originally Posted by pzkpfw
You say "for convenience" and "ad hoc addition". I would say
"correction", "refinement" and "improvement".
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It departs from the essential meaning of what a prime number is.
It adds an artificial restriction to make certain theorems simpler.
'One' clearly would be considered the quintessential prime number
if not for the desire to simplify those theorems. 'One' is a positive
integer which is not the product of other integers. It is the prime
prime number in every way except by the modern definition.
-- Jeff, in Minneapolis