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Prime definitions that exclude 2

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  • shuangtheman
    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. 2. A prime is a positive
    Message 1 of 6 , Aug 7 3:22 PM
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      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.

      2. A prime is a positive integer that cannot be expressed by the product of any of its
      smaller positive integers >1. Implication: if a number has no smaller numbers >1, the
      definition would not apply to that number.

      3. A prime is a positive integer that has only two divisors. Implication: For primes
      greater than 2, only two divisors means we have tested and confirmed that other smaller
      numbers are non-divisors. For the number 2, however, only two divisors means we have
      not tested other smaller numbers as divisors or we have nothing available to test. So, only
      two divisors means opposite things to the real primes versus the number 2. It is sloppy
      logic to use this definition to slip 2 into the class of real primes.


      4. A prime is a positive integer that can be proven to be non-dividable by a smaller
      number >1.

      5. A prime is an odd positive integer that has only two divisors.

      6. The essence of prime is non-dividability by any smaller number >1.

      7. A number is not a prime unless it can be proven to be non-dividable by a smaller
      number >1.

      8. A thing is unique if it is not an inherent part of something else. A prime is a positive
      integer with the property of uniqueness. Such property is not an inherent part of any
      single smaller number. The property of a number is an inherent part of a smaller number
      either because the number is needed for the smaller number to have meaning or its
      property of non-uniqueness can be expressed as a pattern or sums of a single smaller
      number >1. The number 2 is an inherent part of the creation of the number 1, as
      evidenced by the existence of civilizations that had invented only 1 and 2 but no numbers
      beyond 2 and by the absence of civilizations that invented only 1 but not 2. We need 2 to
      invent 1 or for 1 to have any meaning. We need both 1 and 2 in order to invent the
      concept of number. However, we do not need 3 to invent 1 and 2. All numbers are
      inherent in the number 1 as patterns of 1s but the property of uniqueness is not inherent
      in the pattern of 1s. The uniqueness of 11 cannot be expressed as a pattern of 1 that
      would in itself say that 11 is unique but 12 is not. The non-uniqueness of a non-prime
      number is inherently associated with at least one single smaller number. The number 12
      is inherently associated with 3 as a pattern of 3, and this association reveals that 12 is not
      unique.


      Prime has meaning and value only with regard to odd numbers. Why make an exception to
      accommodate one number while disregard the many fundamental differences between this
      one number and the rest? Only left-handed amino acids are relevant to life on earth while
      right-handed amino acids are not. Only odd numbers are relevant to the concept of
      primes while even numbers are not.

      The essence of Prime is about uniqueness and no even numbers can claim to be unique.
      Dividability is a way of measuring uniqueness but is not the essence of primes. A number
      can be non-dividable but still lacks uniqueness, such as 2. The present conventional view
      of 2 as a prime mistook a secondary property (non-dividability) of a primary property
      (uniqueness) as the primary property. Uniqueness cannot exist without something else
      serving as the contrasting background of non-uniqueness. Odd cannot exist without the
      concept of even. 1 cannot exist without 2. Uniqueness is oneness and No numbers could
      be more unique than 1. The uniqueness of 1 demands 1 to be a prime and 2 a non-prime
      by default.

      "317 is a prime, not because we think so, or because our minds are shaped in one way
      rather than another, but *because it is so*, because mathematical reality is built that way."

      --G. H. Hardy, "A Mathematician's Apology"
      Cambridge University Press, 1940.

      I doubt very much that Hardy or any mathematician in his right mind would say the same
      thing about the number 2. 2 may be a prime today but it was not viewed as a prime at
      least once in human history and is most likely not a prime in the objective truth of a
      supernatural reality of God. 317 is a prime no matter how you define prime. But 2 does
      not qualify as a prime in many definitions as I listed above. 317 is an objective prime
      whereas 2 is an artificial prime invented by human conventions of today that will surely be
      proven to be misguided.



      Can any one offer a list of definitions that would include 2 as a prime?
    • jbrennen
      ... I already did, but let me rephrase it in a more concise way... *************************************************************** A natural number X is prime
      Message 2 of 6 , Aug 7 4:18 PM
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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 3 of 6 , Aug 7 5:12 PM
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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 4 of 6 , Aug 7 5:30 PM
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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 5 of 6 , Aug 7 11:05 PM
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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 6 of 6 , Aug 7 11:23 PM
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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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