• ... Then you are at odds with the Fundamental Theorem of Arithmetic. Also, there is a very basic way to define prime -- perhaps the most basic way of all.
Message 1 of 15 , Aug 4, 2006
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--- shuangtheman wrote:
>
> I want to make a conjecture that no definition of
> primes can include 2 to be a prime.

Then you are at odds with the Fundamental Theorem of Arithmetic.

Also, there is a very basic way to define prime -- perhaps the
most basic way of all. Divide the numbers into four categories:

Zero: divides no other number, only itself
Unit: divides every number
Prime: a number P which is not a zero and not a unit, and for
which we can say that if P divides the product of A & B,
then P must divide at least one of A or B
Composite: a number which is not a zero, not a unit, not a prime

This describes the integers perfectly, and without ever bringing
up such "rules" that you seem to find fault with -- it doesn't
use concepts like "less than", nor does it state that a prime is
divisible only by itself and 1 (although you might derive that).

Using this definition, the fact that 2 is prime is equivalent
to the assertion that no even number is a product of two odd
numbers. I'm sure you accept the truth of that statement, right?

Jack
• ... Come up with your own definition of a term that already has a well-established unwavering definition. Don t do that, m kay? ... The Tietze citation looks
Message 2 of 15 , Aug 4, 2006
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--- shuangtheman <shuangtheman@...> wrote:
> I want to

has a well-established unwavering definition.

Don't do that, m'kay?

> While 2 is
> considered a prime today, at one time it was not
> (Tietze 1965, p. 18; Tropfke 1921, p. 96). These
> references are from
> hope that nobody is saying that those people were
> fools.
...
> Tietze, H. "Prime Numbers and Prime Twins." Ch. 1 in
> Famous Problems of Mathematics: Solved and Unsolved
> Mathematics Problems from Antiquity to Modern Times.
> New York: Graylock Press, pp. 1-20, 1965.
>
> Tropfke, J. Geschichte der Elementar-Mathematik, Band
> 1. Berlin, Germany: p. 96, 1921.

The Tietze citation looks like it's to a secondary or tertiary
source therefore no actual primary source for the a definition
that excludes 2 is provided. So your claim is on thin ice.

And of course your "I hope that nobody is saying that those
people were fools", is completely flawed argumentation.
If these guys were simply reporting on the contradictory things
that others had done, then it could quite easily contain the
babbling of loons, yet the authors themselves would not be
fools. Have you never read any Martin Gardner? What we think
of those authors is not relevant, be they gurus or cranks;
only our opinion on the worth of a definition of prime that
excludes 2 is important.

To be honest, my first estimation would be "worthless", but I
could probably be argued down.

Phil

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• I actually think division by zero is not defined. I suggest the definition of primes using set theory: A prime is a natural number, whichs set of divisors has
Message 3 of 15 , Aug 5, 2006
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I actually think division by zero is not defined.

I suggest the definition of primes using set theory: A prime is a natural number, whichs set of divisors has exactly 2 elements.

As zero is no natural number, it falls off. 1 has only one divisor, is therefore no prime. 2 has two divisors, therefore is prime.

Best regards

Michael Paridon

-------- Original-Nachricht --------
Datum: Fri, 04 Aug 2006 23:56:53 -0000
Von: "jbrennen" <jb@...>
Betreff: [PrimeNumbers] Re: Is 2 a prime

> --- shuangtheman wrote:
> >
> > I want to make a conjecture that no definition of
> > primes can include 2 to be a prime.
>
> Then you are at odds with the Fundamental Theorem of Arithmetic.
>
>
> Also, there is a very basic way to define prime -- perhaps the
> most basic way of all. Divide the numbers into four categories:
>
> Zero: divides no other number, only itself
> Unit: divides every number
> Prime: a number P which is not a zero and not a unit, and for
> which we can say that if P divides the product of A & B,
> then P must divide at least one of A or B
> Composite: a number which is not a zero, not a unit, not a prime
>
>
> This describes the integers perfectly, and without ever bringing
> up such "rules" that you seem to find fault with -- it doesn't
> use concepts like "less than", nor does it state that a prime is
> divisible only by itself and 1 (although you might derive that).
>
> Using this definition, the fact that 2 is prime is equivalent
> to the assertion that no even number is a product of two odd
> numbers. I'm sure you accept the truth of that statement, right?
>
>
> Jack
>
>
>
>

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• ... I would suggest that one can define divisibility by zero without needing a definition for division by zero. Say that X is divisible by A if there exists
Message 4 of 15 , Aug 5, 2006
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Dr. Michael Paridon wrote:
> I actually think division by zero is not defined.
>

I would suggest that one can define divisibility by zero without
needing a definition for division by zero. Say that X is divisible by A
if there exists any element B such that X = AB.

Thus you do not need to define exactly which B represents X/A, only
that some such B exists. By this definition, zero is divisible by zero.
Non-zero is not divisible by zero.
• Sorry, but I do not agree. Due to correction: a) I think divisibility is defined for natural numbers only. b) You suggested Say that X is divisible by A if
Message 5 of 15 , Aug 7, 2006
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Sorry, but I do not agree.

Due to correction:

a) I think divisibility is defined for natural numbers only.

b) You suggested "Say that X is divisible by A if there exists any element B such that X = AB."

I think it is "...there exists one and only one distinct element B such that X = AB." Which of course leads to non-definition of division by zero in all cases. As a) does, too.

Best regards

Michael Paridon

-------- Original-Nachricht --------
Datum: Sat, 05 Aug 2006 09:05:57 -0700
Von: Jack Brennen <jb@...>
Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

> Dr. Michael Paridon wrote:
> > I actually think division by zero is not defined.
> >
>
> I would suggest that one can define divisibility by zero without
> needing a definition for division by zero. Say that X is divisible by A
> if there exists any element B such that X = AB.
>
>
> Thus you do not need to define exactly which B represents X/A, only
> that some such B exists. By this definition, zero is divisible by zero.
> Non-zero is not divisible by zero.
>

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• ... With what? Please don t top-post, m kay? ... But Jack provides a definition which works for N / { 0 }. Yes, there exist a handful of simple and convenient
Message 6 of 15 , Aug 7, 2006
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--- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
> Sorry, but I do not agree.

With what?

Fixing:

> Von: Jack Brennen <jb@...>
> > Dr. Michael Paridon wrote:
> > > I actually think division by zero is not defined.
> >
> > I would suggest that one can define divisibility by zero without
> > needing a definition for division by zero. Say that X is divisible by A
> > if there exists any element B such that X = AB.
> >
> >
> > Thus you do not need to define exactly which B represents X/A, only
> > that some such B exists. By this definition, zero is divisible by zero.
> > Non-zero is not divisible by zero.

> Due to correction:
>
> a) I think divisibility is defined for natural numbers only.

But Jack provides a definition which works for N \/ { 0 }.
Yes, there exist a handful of simple and convenient definitions which only work
for natural numbers, but Jack's wording was pedantically correct - one can
provide a definition which does what Jack says it does.

> b) You suggested "Say that X is divisible by A if there exists any element B
> such that X = AB."
>
> I think it is "...there exists one and only one distinct element B such that
> X = AB." Which of course leads to non-definition of division by zero in all
> cases. As a) does, too.

That's one possible definition, yes. If Jack were to rely on his definition of
divisibility in a paper, I feel sure that he would include that definition if
there was any chance of ambiguity.

To be deliberately contrary (shock horror!) I would propose that the simplest
definition of divisibility is one which doesn't mention division at all, it
simply refers to properties of ideals. a|b := (b) \subset (a).

You might enjoy working out the divisibility properties of 0 using this
definition.

Phil

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• ... Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT) Von: Phil Carmody An: primenumbers@yahoogroups.com Betreff: Re: [PrimeNumbers] Re: Is
Message 7 of 15 , Aug 7, 2006
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-------- Original-Nachricht --------
Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT)
Von: Phil Carmody <thefatphil@...>
Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

> --- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
> > Sorry, but I do not agree.
>
> With what?
>
>
> Fixing:
>
> > Von: Jack Brennen <jb@...>
> > > Dr. Michael Paridon wrote:
> > > > I actually think division by zero is not defined.
> > >
> > > I would suggest that one can define divisibility by zero without
> > > needing a definition for division by zero. Say that X is divisible by
> A
> > > if there exists any element B such that X = AB.
> > >
> > >
> > > Thus you do not need to define exactly which B represents X/A, only
> > > that some such B exists. By this definition, zero is divisible by
> zero.
> > > Non-zero is not divisible by zero.
>
> > Due to correction:
> >
> > a) I think divisibility is defined for natural numbers only.
>
> But Jack provides a definition which works for N \/ { 0 }.
> Yes, there exist a handful of simple and convenient definitions which only
> work
> for natural numbers, but Jack's wording was pedantically correct - one can
> provide a definition which does what Jack says it does.
>
> > b) You suggested "Say that X is divisible by A if there exists any
> element B
> > such that X = AB."
> >
> > I think it is "...there exists one and only one distinct element B such
> that
> > X = AB." Which of course leads to non-definition of division by zero in
> all
> > cases. As a) does, too.
>
> That's one possible definition, yes. If Jack were to rely on his
> definition of
> divisibility in a paper, I feel sure that he would include that definition
> if
> there was any chance of ambiguity.
>
> To be deliberately contrary (shock horror!) I would propose that the
> simplest
> definition of divisibility is one which doesn't mention division at all,
> it
> simply refers to properties of ideals. a|b := (b) \subset (a).
>
> You might enjoy working out the divisibility properties of 0 using this
> definition.
>
> Phil
>
> () ASCII ribbon campaign () Hopeless ribbon campaign
> /\ against HTML mail /\ against gratuitous bloodshed
>
> [stolen with permission from Daniel B. Cristofani]

Sorry, if I did not post right. I was not about to cause any inconvenience.

I did not say Jack's definition is not possible. I think is not the definition mostly used, as usally division by zero is not defined. As well as 0^0, if I remember right. Jack's definition leads to an agreeable result, I admit.

Best regards

Michael Paridon

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• ... It depends on the area people work in. For a combinatorian, the equality 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number of
Message 8 of 15 , Aug 7, 2006
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> Sorry, if I did not post right. I was not about to cause any
> inconvenience.
>
> I did not say Jack's definition is not possible. I think is not the
> definition mostly used, as usally division by zero is not defined. As
> well as 0^0, if I remember right. Jack's definition leads to an

It depends on the area people work in. For a combinatorian, the equality
0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
of functions from empty set to empty set [*]. Moreover, things like
binomial theorem also work nice with this extension; it allows you to
evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
simply 0^n (ok, I admit, this is just a contrived academic example).

Likewise, if you work in the area of foundations of mathematics, defining
divisibility using the operation of division is a bit more complicated
than using the straight existential-quantifer with multiplication (just
like Jack did); for the division is only a derived operation in e.g. Peano
Arithmetics and one needs to prove its well-definedness (and possibly some
other properties) first.

On the other hand, an analyst would probably bop you over the head
if he saw 0^0 :-)

Peter

[*] This works even in the much more general framework of set-theory --
If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
to be the cardinality of the set A^B which is the set of all functions
from B to A. If the sets A and B are finite, the cardinal exponentation
agrees with the usual exponentation of natural numbers.

--
[Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
• So I thank everybody very much for explanation. Mathematics has more hidden beauties I will ever learn, and as I am not in a professional way dealing with
Message 9 of 15 , Aug 7, 2006
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So I thank everybody very much for explanation. Mathematics has more hidden
beauties I will ever learn, and as I am not in a professional way dealing
with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

Michael

----- Original Message -----
From: "Peter Kosinar" <goober@...>
To: "Dr. Michael Paridon" <dr.m.paridon@...>
Sent: Monday, August 07, 2006 4:19 PM
Subject: Re: [PrimeNumbers] Re: Is 2 a prime

> > Sorry, if I did not post right. I was not about to cause any
> > inconvenience.
> >
> > I did not say Jack's definition is not possible. I think is not the
> > definition mostly used, as usally division by zero is not defined. As
> > well as 0^0, if I remember right. Jack's definition leads to an
> > agreeable result, I admit.
>
> It depends on the area people work in. For a combinatorian, the equality
> 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
> of functions from empty set to empty set [*]. Moreover, things like
> binomial theorem also work nice with this extension; it allows you to
> evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
> simply 0^n (ok, I admit, this is just a contrived academic example).
>
> Likewise, if you work in the area of foundations of mathematics, defining
> divisibility using the operation of division is a bit more complicated
> than using the straight existential-quantifer with multiplication (just
> like Jack did); for the division is only a derived operation in e.g. Peano
> Arithmetics and one needs to prove its well-definedness (and possibly some
> other properties) first.
>
> On the other hand, an analyst would probably bop you over the head
> if he saw 0^0 :-)
>
> Peter
>
> [*] This works even in the much more general framework of set-theory --
> If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
> to be the cardinality of the set A^B which is the set of all functions
> from B to A. If the sets A and B are finite, the cardinal exponentation
> agrees with the usual exponentation of natural numbers.
>
> --
> [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
>
>
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> Diese E-Mail wurde vom NOD32 antivirus system geprüft
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>
• ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
Message 10 of 15 , Aug 7, 2006
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Dr. Michael Paridon wrote:
> So I thank everybody very much for explanation. Mathematics has more hidden
> beauties I will ever learn, and as I am not in a professional way dealing
> with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction.