## Re: [PrimeNumbers] Re: Prime definitions that exclude 2

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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
Message 1 of 6 , Aug 7, 2006
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 2 of 6 , Aug 7, 2006
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 3 of 6 , Aug 7, 2006
--- 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 4 of 6 , Aug 7, 2006
--- 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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