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Re: This has to be flaky

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  • David Broadhurst <d.broadhurst@open.ac.u
    Neither Bookfinder nor Amazon could find a sellable copy of Jackson: http://www.amazon.co.uk/exec/obidos/ASIN/0710079982 It was fine little book, since Walter
    Message 1 of 14 , Jan 4, 2003
      Neither Bookfinder nor Amazon could find a sellable copy of Jackson:

      http://www.amazon.co.uk/exec/obidos/ASIN/0710079982

      It was fine little book, since Walter Ledermann, the series editor,
      demanded the highest standards of pedagogy, aimed at good high-school
      or first year undergrad students.

      The 1975 price was 1.5 GPB ~ $2.

      David
    • Jon Perry
      The proof I have is much simpler. sorry about the Hardy credit - it s in his book, and I thought I had read something somewhere which said it was Hardy s. This
      Message 2 of 14 , Jan 4, 2003
        The proof I have is much simpler. sorry about the Hardy credit - it's in his
        book, and I thought I had read something somewhere which said it was
        Hardy's.

        This is from Number Theory by Naoki Sato, which is freely available as a
        PDF.

        There exists 'a' such that a^2=-1modp

        Consider the set of integer ax-y, x,y integers, o<=x<sqrt(p).

        The number of possible pairs (x,y) is then
        [floor(sqrt(p))+1]^2>[sqrt(p)]^2=p

        So, by the pigeonhole principle, there exist 0<=x1,x2,y1,y2<sqrt(p) such
        that

        ax1-y1 = ax2 - y2 modp.

        Let x=x1-x2 and y=y1-y2. At least one of x and y is non-zero.

        Thus x^2 + y^2 is a multiple of p, and 0<x^2+y^2<sqrt(p)^2+sqrt(p)^2=2p,

        Hence x^2+y^2=p.

        Jon Perry
        perry@...
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      • Phil Carmody
        ... Your original wording offers you a get out of jail card - it probably was proved by Hardy, whether any other number of mathematicians proved it before that
        Message 3 of 14 , Jan 4, 2003
          --- Jon Perry <perry@...> wrote:
          > The proof I have is much simpler. sorry about the Hardy credit - it's in his
          > book, and I thought I had read something somewhere which said it was
          > Hardy's.

          Your original wording offers you a get out of jail card - it probably was
          proved by Hardy, whether any other number of mathematicians proved it before
          that is irrelevant.

          > This is from Number Theory by Naoki Sato, which is freely available as a
          > PDF.
          >
          > There exists 'a' such that a^2=-1modp

          There may exist 2.

          > Consider the set of integer ax-y, x,y integers, o<=x<sqrt(p).

          I assume you mean 0 not o.

          > The number of possible pairs (x,y) is then

          infinite, as there's no restriction on y. I assume you meant 0<=x,y<sqrt(p)

          > [floor(sqrt(p))+1]^2>[sqrt(p)]^2=p
          >
          > So, by the pigeonhole principle, there exist 0<=x1,x2,y1,y2<sqrt(p) such
          > that
          >
          > ax1-y1 = ax2 - y2 modp.
          >
          > Let x=x1-x2 and y=y1-y2. At least one of x and y is non-zero.

          The pidgeon hole principle cannot immediately on its own guarantee that
          both x and y are non-zero, but ax-y == 0 (mod p) implies that if one is
          zero, the other is too, a contradiction. Therefore you conclude that both
          are non-zero.

          > Thus x^2 + y^2 is a multiple of p, and 0<x^2+y^2<sqrt(p)^2+sqrt(p)^2=2p,
          >
          > Hence x^2+y^2=p.

          I like it. It's less constructive than the Fermat version, but nonetheless,
          you can't doubt its verity. I quite like pigeonhole proofs, often they're
          very elegant, this is no exception.

          Phil


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        • David Broadhurst <d.broadhurst@open.ac.u
          Jackson taught Fermat s original proof by descent for an excellent pedagogic reason: to prepare the student for Lagrange s tour de force, by descent, in
          Message 4 of 14 , Jan 4, 2003
            Jackson taught Fermat's original proof by descent for an excellent
            pedagogic reason: to prepare the student for Lagrange's
            tour de force, by descent, in proving

            Theorem [Lagrange]: Every natural number can be expressed
            as a sum of four integer squares

            whose proof proceeds along the same lines as the one I sketched.

            Short proofs are not always the most instructive.
          • Jon Perry
            From: http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html Fermat described his method of infinite descent and gave an example on how it could
            Message 5 of 14 , Jan 5, 2003
              From:

              http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html

              'Fermat described his method of infinite descent and gave an example on how
              it could be used to prove that every prime of the form 4k + 1 could be
              written as the sum of two squares. For suppose some number of the form 4k +
              1 could not be written as the sum of two squares. Then there is a smaller
              number of the form 4k + 1 which cannot be written as the sum of two squares.
              Continuing the argument will lead to a contradiction. What Fermat failed to
              explain in this letter is how the smaller number is constructed from the
              larger. One assumes that Fermat did know how to make this step but again his
              failure to disclose the method made mathematicians lose interest. It was not
              until Euler took up these problems that the missing steps were filled in.'

              Jon Perry
              perry@...
              http://www.users.globalnet.co.uk/~perry/maths/
              http://www.users.globalnet.co.uk/~perry/DIVMenu/
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            • David Broadhurst <d.broadhurst@open.ac.u
              Thanks Jon for that St Andrews link! Just goes to show that pedagogy is the enemy of good history. David
              Message 6 of 14 , Jan 5, 2003
                Thanks Jon for that St Andrews link!
                Just goes to show that pedagogy is the enemy of good history.
                David
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