• 1. A prime is a positive integer that cannot be expressed by the even number of sums of any single number except 1 and itself. 2. A prime is a positive
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1. A prime is a positive integer that cannot be expressed by the even number of sums of
any single number except 1 and itself.

2. A prime is a positive integer that cannot be expressed by the product of any of its
smaller positive integers >1. Implication: if a number has no smaller numbers >1, the
definition would not apply to that number.

3. A prime is a positive integer that has only two divisors. Implication: For primes
greater than 2, only two divisors means we have tested and confirmed that other smaller
numbers are non-divisors. For the number 2, however, only two divisors means we have
not tested other smaller numbers as divisors or we have nothing available to test. So, only
two divisors means opposite things to the real primes versus the number 2. It is sloppy
logic to use this definition to slip 2 into the class of real primes.

4. A prime is a positive integer that can be proven to be non-dividable by a smaller
number >1.

5. A prime is an odd positive integer that has only two divisors.

6. The essence of prime is non-dividability by any smaller number >1.

7. A number is not a prime unless it can be proven to be non-dividable by a smaller
number >1.

8. A thing is unique if it is not an inherent part of something else. A prime is a positive
integer with the property of uniqueness. Such property is not an inherent part of any
single smaller number. The property of a number is an inherent part of a smaller number
either because the number is needed for the smaller number to have meaning or its
property of non-uniqueness can be expressed as a pattern or sums of a single smaller
number >1. The number 2 is an inherent part of the creation of the number 1, as
evidenced by the existence of civilizations that had invented only 1 and 2 but no numbers
beyond 2 and by the absence of civilizations that invented only 1 but not 2. We need 2 to
invent 1 or for 1 to have any meaning. We need both 1 and 2 in order to invent the
concept of number. However, we do not need 3 to invent 1 and 2. All numbers are
inherent in the number 1 as patterns of 1s but the property of uniqueness is not inherent
in the pattern of 1s. The uniqueness of 11 cannot be expressed as a pattern of 1 that
would in itself say that 11 is unique but 12 is not. The non-uniqueness of a non-prime
number is inherently associated with at least one single smaller number. The number 12
is inherently associated with 3 as a pattern of 3, and this association reveals that 12 is not
unique.

Prime has meaning and value only with regard to odd numbers. Why make an exception to
accommodate one number while disregard the many fundamental differences between this
one number and the rest? Only left-handed amino acids are relevant to life on earth while
right-handed amino acids are not. Only odd numbers are relevant to the concept of
primes while even numbers are not.

The essence of Prime is about uniqueness and no even numbers can claim to be unique.
Dividability is a way of measuring uniqueness but is not the essence of primes. A number
can be non-dividable but still lacks uniqueness, such as 2. The present conventional view
of 2 as a prime mistook a secondary property (non-dividability) of a primary property
(uniqueness) as the primary property. Uniqueness cannot exist without something else
serving as the contrasting background of non-uniqueness. Odd cannot exist without the
concept of even. 1 cannot exist without 2. Uniqueness is oneness and No numbers could
be more unique than 1. The uniqueness of 1 demands 1 to be a prime and 2 a non-prime
by default.

"317 is a prime, not because we think so, or because our minds are shaped in one way
rather than another, but *because it is so*, because mathematical reality is built that way."

--G. H. Hardy, "A Mathematician's Apology"
Cambridge University Press, 1940.

I doubt very much that Hardy or any mathematician in his right mind would say the same
thing about the number 2. 2 may be a prime today but it was not viewed as a prime at
least once in human history and is most likely not a prime in the objective truth of a
supernatural reality of God. 317 is a prime no matter how you define prime. But 2 does
not qualify as a prime in many definitions as I listed above. 317 is an objective prime
whereas 2 is an artificial prime invented by human conventions of today that will surely be
proven to be misguided.

Can any one offer a list of definitions that would include 2 as a prime?
• ... I already did, but let me rephrase it in a more concise way... *************************************************************** A natural number X is prime
Message 2 of 6 , Aug 7, 2006
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--- shuangtheman wrote:
>
> Can any one offer a list of definitions that would include
> 2 as a prime?

I already did, but let me rephrase it in a more concise way...

***************************************************************
A natural number X is prime if the set of natural numbers not
divisible by X is non-empty and is closed under multiplication.
***************************************************************

So it says two things: divisibility by X is not an inherent
property (so 1 is excluded), and divisibility by X cannot be
"created out of non-divisibility" -- the only way to have a
product divisible by X is to have one of the multiplicands
divisible by X.

This precisely describes the prime numbers. Explain to me how
this definition is faulty for the number 2, but not for any
odd primes.
• I can also give you a definition that will include 2 as a prime: A prime is a positive integer that has only two divisors, regardless whether it has smaller
Message 3 of 6 , Aug 7, 2006
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I can also give you a definition that will include 2
as a prime:
A prime is a positive integer that has only two
divisors, regardless whether it has smaller number
that can be tested for divisibility. In this case,
even though 2 is much different from other odd primes,
we artificially treated it as the same as other
primes.

I am not a math specialist and so I dont quite follow
your definition. But to have a way of difining 2 to
be a prime cannot refute the fact that 2 is very
different from all other primes. We can easily
exclude 2 as a prime but we cannot do the same with
317. So there is no objective standard on 2 as there
is on 317. Thus ultimately, it is subjective human
conventions that decides wether 2 is a prime. Humans
should just be honest and say flatly that 2 is a prime
not because it is like 317 but because we want it to
be. We should just be as honest as we treated 1. We
say 1 is not a prime not because it is not but because
we dont want it to be.

Below is a honest quot from math world on prime
numbers.

As more simply noted by Derbyshire (2004, p. 33), "2
pays its way [as a prime] on balance; 1 doesn't."
Derbyshire, J. Prime Obsession: Bernhard Riemann and
the Greatest Unsolved Problem in Mathematics. New
York: Penguin, 2004.

So, for anyone to insist that 2 is a prime based on
objective truth, the same as 317, is not really being
honest. To attemp to justify our artificical
conventions by cleverly formulating a seemingly
objective definition that does include 2 is not being
totally honest. The honest thing to do is to say that
2 could easily be a prime or a non-prime, but it suits
our purpose better if it is treated as a prime. Just
like 1 could easily be a prime or non-prime, but it
suits our purpose better if it is treated as a
non-prime. But the purpose of today may not be
relevant to the objective truth.

--- jbrennen <jb@...> wrote:

> --- shuangtheman wrote:
> >
> > Can any one offer a list of definitions that would
> include
> > 2 as a prime?
>
> I already did, but let me rephrase it in a more
> concise way...
>
>
***************************************************************
> A natural number X is prime if the set of natural
> numbers not
> divisible by X is non-empty and is closed under
> multiplication.
>
***************************************************************
>
> So it says two things: divisibility by X is not an
> inherent
> property (so 1 is excluded), and divisibility by X
> cannot be
> "created out of non-divisibility" -- the only way to
> have a
> product divisible by X is to have one of the
> multiplicands
> divisible by X.
>
> This precisely describes the prime numbers. Explain
> to me how
> this definition is faulty for the number 2, but not
> for any
> odd primes.
>
>
>
>
>
>

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• Following a list of primes that starts with 2, the author Derbyshire wrote: “At this point, someone usually objects that 1 is not included in this or any
Message 4 of 6 , Aug 7, 2006
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Following a list of primes that starts with 2, the
author Derbyshire wrote: At this point, someone
usually objects that 1 is not included in this or any
other list of primes. It fits the definition, doesnt
it? Well, yes, strictly speaking, it does, and if you
want to be a barrack-rood lawyer about it, you can
write in a 1 at the start of the list for your own
satisfacion. Including 1 in the primes, however, is a
major nuisance, and modern mathematicians just dont,
by common agreement. (The last mathematician of any
importance who did seems to have been Henri Lebesgue,
in 1899.) Even including 2 is a nuisance, actually.
Countless theorems begin with, Let p be any odd
prime. However, 2 pays its way on balance; 1
doesnt, so we just leave it out.

from Derbyshire, J. Prime Obsession: Bernhard Riemann
and the Greatest Unsolved Problem in Mathematics. New
York: Penguin, 2004. Page 33.

So the above quote proves my point that 2 is treated
as a prime today the same way as 1 is not, by human
agreement rather than objective truth. 2 as prime
serves us better and so lets call it a prime. 1 as a
prime does serve us as well so lets ignore it.
Clearly serving us is not the same as serving God. If
God is uniqueness there is no way He would consider
uniqueness and in turn the related concept of oneness
to be anything other than a prime. There is no way He
would treat a follower of uniqueness/oneness, such as
2 following 1 or even following odd, to be a prime.

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• ... It does not. You re right that it s human agreement, but any selection of axioms (and postulates) and inference rules, for example is a human agreement. To
Message 5 of 6 , Aug 7, 2006
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--- Shi Huang <shuangtheman@...> wrote:
> Following a list of primes that starts with 2, the
> author Derbyshire wrote: At this point, someone
> usually objects that 1 is not included in this or any
> other list of primes. It fits the definition, doesnt
> it? Well, yes, strictly speaking, it does, and if you
> want to be a barrack-rood lawyer about it, you can
> write in a 1 at the start of the list for your own
> satisfacion. Including 1 in the primes, however, is a
> major nuisance, and modern mathematicians just dont,
> by common agreement. (The last mathematician of any
> importance who did seems to have been Henri Lebesgue,
> in 1899.) Even including 2 is a nuisance, actually.
> Countless theorems begin with, Let p be any odd
> prime. However, 2 pays its way on balance; 1
> doesnt, so we just leave it out.
>
> from Derbyshire, J. Prime Obsession: Bernhard Riemann
> and the Greatest Unsolved Problem in Mathematics. New
> York: Penguin, 2004. Page 33.
>
> So the above quote proves my point that 2 is treated
> as a prime today the same way as 1 is not, by human
> agreement rather than objective truth.

It does not.

You're right that it's human agreement, but any selection
of axioms (and postulates) and inference rules, for example
is a human agreement. To think otherwise shows a complete
disregard for how the foundations of modern mathematics are
defined.

However, there are fundamental differences between the issues
that '1' causes, and the issues that '2' causes which make them
not comparable.

> 2 as prime
> serves us better and so lets call it a prime. 1 as a
> prime does serve us as well so lets ignore it.

I assume that should read "doesn't". But it's still a gross
misrepresentation of the truth. Having a unit as a prime messes
up /almost everything/.

> Clearly serving us is not the same as serving God.

Obviously. Ockham's razor indicates that there's no need to have
brought up the latter at all, and persuades us that the simplest
set of rules is usually the better one. Having a unit as a prime
complicates almost every otherwise simple rule we have, and
therefore is unwarranted, and unwanted.

It appears that you don't understand _why_ 2 causes the problems
that it does in the situations where one needs to say "an odd prime".
It's usually not its primeness that causes the problem, but it's
_size_. It is the only prime for which x == -x (mod p) for all x,
which messes up assumptions about orders (see carmichael's lambda,
for example). Lack of divisibility by 2 messes things up too,
but lack of divisibility by 3 messes things up in elliptic curves
over GF(3^n), and there's no temptation to not call 3 a prime -
it wasn't the _primeness_ of 2 that was the problem, merely the fact
that 2 occured as a multiplier that one wanted to invert.

Coupled to this, it appears that you don't understand why having
1 causes the problems that it does too. The fact that it is a unit
messes up practically everything it touches.

The reason why units have been isolated in their own special
category is for a very simple reason - they behave fudamentally
differently from the other members of the multiplicative group.

To group them all together and then to have to separate them again
almost everywhere provides the mathematician with no perceptable
gain, and plenty of pain.

I notice that precisely _no_ sources for primes to concretely be
defined by a mathematician to exclude 2 have been cited yet, which
reinforces my previous post on the subject.

Phil

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• ... How is 2 expressible as an even number of sums of a single number that isn t 1 or 2? It can t. So your definition of prime that excludes 2 includes 2. Does
Message 6 of 6 , Aug 7, 2006
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--- shuangtheman <shuangtheman@...> wrote:
> 1. A prime is a positive integer that cannot be expressed by the even number
> of sums of any single number except 1 and itself.

How is 2 expressible as an even number of sums of a single number that isn't 1
or 2? It can't. So your definition of prime that excludes 2 includes 2.

Does this depend on what the meaning on of 'is' is, or something?

I think the latter, and I recommend retreating.

Phil

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