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

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  • 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 1 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 2 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 3 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 4 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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