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18273Re: [PrimeNumbers] Re: Prime definitions that exclude 2

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  • Phil Carmody
    Aug 7, 2006
      --- 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

      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.


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