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Re: Prime definitions that exclude 2

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  • jbrennen
    ... I already did, but let me rephrase it in a more concise way... *************************************************************** A natural number X is prime
    Message 1 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.
    • Shi Huang
      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 2 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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      • Shi Huang
        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 3 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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        • Phil Carmody
          ... 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 4 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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          • Phil Carmody
            ... 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 5 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?

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

              I think the latter, and I recommend retreating.

              Phil


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