- --- shuangtheman wrote:
>

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

> Can any one offer a list of definitions that would include

> 2 as a prime?

***************************************************************

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 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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http://mail.yahoo.com - 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, doesnt

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 dont,

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

doesnt, 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 lets call it a prime. 1 as a

prime does serve us as well so lets 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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http://mail.yahoo.com - --- Shi Huang <shuangtheman@...> wrote:
> Following a list of primes that starts with 2, the

It does not.

> 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, doesnt

> 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 dont,

> 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

> doesnt, 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.

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

I assume that should read "doesn't". But it's still a gross

> serves us better and so lets call it a prime. 1 as a

> prime does serve us as well so lets ignore it.

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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http://mail.yahoo.com - --- shuangtheman <shuangtheman@...> wrote:
> 1. A prime is a positive integer that cannot be expressed by the even number

How is 2 expressible as an even number of sums of a single number that isn't 1

> of sums of any single number except 1 and itself.

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?

Or have your just shot yourself in the foot _really_ badly.

I think the latter, and I recommend retreating.

Phil

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