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Re: [PrimeNumbers] Is 2 a prime

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  • Thomas Hadley
    ... A good warning, Paul. In case this reference is confusing to some, here s an explanation. This is from an article by Kurt Salzinger of the American
    Message 1 of 15 , Aug 4 3:25 PM
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      primenumbers@yahoogroups.com wrote on 08/04/2006 04:49:35 PM:

      > On Fri, 2006-08-04 at 22:27, shuangtheman wrote:
      > > I want to make a conjecture that no definition of
      > > primes can include 2 to be a prime. A definition
      >
      > Huumpty-dumpty alert!
      >
      > Paul

      A good warning, Paul. In case this reference is confusing
      to some, here's an explanation. This is from an article by
      Kurt Salzinger of the American Psychological Association:

      Thus we use everyday words, following the Humpty Dumpty
      prescription in which he contends that the meaning of a
      given word is a question of ?which is to be master . . .
      When I use a word, it means just what I choose it to
      mean - - neither more nor less.? The problem is that
      while this approach may be amusing in a children?s book,
      it does not help in communicating with the wide world
      outside.


      Tom Hadley

      [Non-text portions of this message have been removed]
    • shuangtheman
      Simon, I like your comments. Indeed, the primes may represent a code for the ETI or even the supernatural God to communicate with humans. But mere awareness
      Message 2 of 15 , Aug 4 4:20 PM
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        Simon,

        I like your comments. Indeed, the primes may represent a code for the ETI or even the
        supernatural God to communicate with humans. But mere awareness of primes is rather
        primitive and the series of primes, 3, 5, 7, 11,... may be encoding a message that has
        nothing to do with numbers. Until humans figure out why 2 is or is not a prime, the
        supernatural would have no interest in contacting our lowly intelligence. There may be a
        message encoded in the definition of primes or the proper way of defining and generating
        primes. When we become intelligent enough to figure out that message, we would have
        known whether there is or is not a supernatural world out there. We would have known
        whether to consider 2 a prime based on objective truth rather than arbitrary human
        convenience. Most first rate mathematicians believe a supernatural Platonic world of
        objective mathematical truth. There must be a truth on the primality of 2. Whatever that
        truth may be, it is clear that we humans have yet to find it since we are presently calling
        the shots on 2 based on our own convenience of playing
        some number theory games.


        --- Simon <4_groups@...> wrote:

        > Carl Sagan, in the novel "Contact", allows that
        > prime numbers are odd integers, as opposed
        > to even integers, which I believe was delivered by
        > the heroine Ellie Arroway, who is the head
        > of the ARGUS project. I just stumbled across this
        > last night while reading the novel, so it is
        > there.
        >
        > This project ultimately found an ETI presence
        > emanating communication by complex radio
        > signals from the star Vega, in Lyra.
        >
        > Go figure, Paul Leyland!
        >
        > Maybe this is why we aren't finding ETI; we're using
        > 2 as a prime number!
        >
        > I, too, admit that it is puzzling why number theory
        > often proceeds forward based an
        > exception. I wish that Paul Leyland would be a bit
        > more constructive, and outline the
        > implications that a change in definition would have
        > on all the number theory. However.
        > perhaps laziness, or a lack of hubris delivers his
        > above comment.
        >
        > Thoughtfully,
        > Simon
        >
        >
        >
        >
      • Chris Caldwell
        On Behalf Of shuangtheman ... The objective truth is simple and obvious (for integers): 2 is prime. Those that can not understand such a trivial definition
        Message 3 of 15 , Aug 4 4:47 PM
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          On Behalf Of shuangtheman
          > Subject: [PrimeNumbers] Re:Is 2 a prime
          > ... There must be a truth on the primality of 2.
          > Whatever that truth may be,

          The objective truth is simple and obvious (for integers):
          2 is prime. Those that can not understand such a trivial
          definition should not admit so as loudly and with as
          many words as they often do.

          CC
        • jbrennen
          ... 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 4 of 15 , Aug 4 4:56 PM
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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
          • Phil Carmody
            ... 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 5 of 15 , Aug 4 11:10 PM
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              --- shuangtheman <shuangtheman@...> wrote:
              > I want to

              Come up with your own definition of a term that already
              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
              > http://mathworld.wolfram.com/PrimeNumber.html. I
              > 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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            • Dr. Michael Paridon
              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 6 of 15 , Aug 5 2:13 AM
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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@...>
                An: primenumbers@yahoogroups.com
                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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              • Jack Brennen
                ... 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 7 of 15 , Aug 5 9:05 AM
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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.
                • Dr. Michael Paridon
                  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 8 of 15 , Aug 7 4:41 AM
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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@...>
                    An: primenumbers@yahoogroups.com
                    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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                  • Phil Carmody
                    ... 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 9 of 15 , Aug 7 5:30 AM
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                      --- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
                      > Sorry, but I do not agree.

                      With what?

                      Please don't top-post, m'kay?

                      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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                    • Dr. Michael Paridon
                      ... Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT) Von: Phil Carmody An: primenumbers@yahoogroups.com Betreff: Re: [PrimeNumbers] Re: Is
                      Message 10 of 15 , Aug 7 6:07 AM
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                        -------- Original-Nachricht --------
                        Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT)
                        Von: Phil Carmody <thefatphil@...>
                        An: primenumbers@yahoogroups.com
                        Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

                        > --- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
                        > > Sorry, but I do not agree.
                        >
                        > With what?
                        >
                        > Please don't top-post, m'kay?
                        >
                        > 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
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                        > [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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                      • Peter Kosinar
                        ... 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 11 of 15 , Aug 7 7:19 AM
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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
                          > 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
                        • Dr. Michael Paridon
                          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 12 of 15 , Aug 7 8:50 AM
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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.

                            Best regrads

                            Michael


                            ----- Original Message -----
                            From: "Peter Kosinar" <goober@...>
                            To: "Dr. Michael Paridon" <dr.m.paridon@...>
                            Cc: <primenumbers@yahoogroups.com>
                            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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                          • Alan McFarlane
                            ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
                            Message 13 of 15 , Aug 7 9:36 AM
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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.

                              > Best regrads
                              >
                              > Michael
                              >
                              >
                              > ----- Original Message -----
                              [snip]


                              --
                              Alan
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