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RE: [PrimeNumbers] New primers

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  • Jon Perry
    Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
    Message 1 of 11 , Jan 8, 2002
      Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry
      http://www.users.globalnet.co.uk/~perry/maths
      BrainBench MVP for HTML and JavaScript
      http://www.brainbench.com


      -----Original Message-----
      From: Phil Carmody [mailto:fatphil@...]
      Sent: 08 January 2002 11:09
      To: primenumbers@yahoogroups.com
      Subject: Re: [PrimeNumbers] New primers


      On Tue, 08 January 2002, "paulmillscv" wrote:
      > Prove that if gcd(a,b) = d, then gcd( |a-b|,min(a,b)) = d. where
      > || is the modulus function, and min(a,b) is the min function.

      That's a bit noisy - gcd is symmetric, so gcd(a,b)=gcd(b,a). Therefore you
      can simply assume WLOG[*] a>=b, and your expression becomes, with no need
      for abs or min,
      gcd(a,b)=d -> gcd(a-b, b)=d

      The proof is almost not deserving of the word, as
      d|a & d|b -> d|(a-b),
      d|(a-b) & d|b -> d|a

      Phil
      [* WLOG = Without Loss Of Generality]





      Don't be fooled, CRC Press are _not_ the good guys.
      They've taken Wolfram's money - _don't_ give them yours.
      http://mathworld.wolfram.com/erics_commentary.html


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    • Paul Leyland
      ... Oh no it isn t! As zero divided by any integer d gives a quotient of zero and a remainder of zero gcd(0,d)=d Paul
      Message 2 of 11 , Jan 8, 2002
        > From: Jon Perry [mailto:perry@...]

        > Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

        Oh no it isn't!

        As zero divided by any integer d gives a quotient of zero and a
        remainder of zero gcd(0,d)=d


        Paul
      • Jon Perry
        An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1), and 0/0 does not equal 0. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
        Message 3 of 11 , Jan 8, 2002
          An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1), and 0/0 does
          not equal 0.

          Jon Perry
          perry@...
          http://www.users.globalnet.co.uk/~perry
          http://www.users.globalnet.co.uk/~perry/maths
          BrainBench MVP for HTML and JavaScript
          http://www.brainbench.com


          -----Original Message-----
          From: Paul Leyland [mailto:pleyland@...]
          Sent: 08 January 2002 18:11
          To: Jon Perry; primenumbers@yahoogroups.com
          Subject: RE: [PrimeNumbers] New primers



          > From: Jon Perry [mailto:perry@...]

          > Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

          Oh no it isn't!

          As zero divided by any integer d gives a quotient of zero and a
          remainder of zero gcd(0,d)=d


          Paul


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        • Jon Perry
          What I meant was gcd(0,p)=p (which is the fishy result) Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
          Message 4 of 11 , Jan 8, 2002
            What I meant was gcd(0,p)=p (which is the fishy result)

            Jon Perry
            perry@...
            http://www.users.globalnet.co.uk/~perry
            http://www.users.globalnet.co.uk/~perry/maths
            BrainBench MVP for HTML and JavaScript
            http://www.brainbench.com


            -----Original Message-----
            From: Jon Perry [mailto:perry@...]
            Sent: 08 January 2002 18:16
            To: primenumbers@yahoogroups.com
            Subject: RE: [PrimeNumbers] New primers


            An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1), and 0/0 does
            not equal 0.

            Jon Perry
            perry@...
            http://www.users.globalnet.co.uk/~perry
            http://www.users.globalnet.co.uk/~perry/maths
            BrainBench MVP for HTML and JavaScript
            http://www.brainbench.com


            -----Original Message-----
            From: Paul Leyland [mailto:pleyland@...]
            Sent: 08 January 2002 18:11
            To: Jon Perry; primenumbers@yahoogroups.com
            Subject: RE: [PrimeNumbers] New primers



            > From: Jon Perry [mailto:perry@...]

            > Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

            Oh no it isn't!

            As zero divided by any integer d gives a quotient of zero and a
            remainder of zero gcd(0,d)=d


            Paul


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          • Paul Leyland
            ... You are quite correct: it is an arbitrary definition, but it s one which is convenient and almost universally accepted in the field of mathematics. Note
            Message 5 of 11 , Jan 8, 2002
              > An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1),
              > and 0/0 does
              > not equal 0.

              You are quite correct: it is an arbitrary definition, but it's one which
              is convenient and almost universally accepted in the field of
              mathematics.

              Note the primality of 0 and 1 has nothing to do with the definition of
              gcd(0,d) and, indeed, neither is prime: again by a definition which is
              both useful and (almost) ubiquitous.

              0/0 is undefined. It's defined to be undefined 8-)


              Paul
            • Jon Perry
              I think 0/0 should be defined as 0. 0 multiplied by any non-zero number gives 0, and I don t see why multiplying it by infinity should be any different. If
              Message 6 of 11 , Jan 8, 2002
                I think 0/0 should be defined as 0. 0 multiplied by any non-zero number
                gives 0, and I don't see why multiplying it by infinity should be any
                different.

                If this is not so, then I could take my zero Euro's, and divide them into my
                zero people, and end up extremely wealthy.

                Jon Perry
                perry@...
                http://www.users.globalnet.co.uk/~perry
                http://www.users.globalnet.co.uk/~perry/maths
                BrainBench MVP for HTML and JavaScript
                http://www.brainbench.com


                -----Original Message-----
                From: Paul Leyland [mailto:pleyland@...]
                Sent: 08 January 2002 18:22
                To: Jon Perry; primenumbers@yahoogroups.com
                Subject: RE: [PrimeNumbers] New primers


                > An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1),
                > and 0/0 does
                > not equal 0.

                You are quite correct: it is an arbitrary definition, but it's one which
                is convenient and almost universally accepted in the field of
                mathematics.

                Note the primality of 0 and 1 has nothing to do with the definition of
                gcd(0,d) and, indeed, neither is prime: again by a definition which is
                both useful and (almost) ubiquitous.

                0/0 is undefined. It's defined to be undefined 8-)


                Paul
              • djbroadhurst
                ? gcd(0,137) %1 = 137 integer g is a divisor of integer f if and only there is an integer g such that f=g*h g=137 is a divisor of f=137 because there is an
                Message 7 of 11 , Jan 8, 2002
                  ? gcd(0,137)
                  %1 = 137

                  integer g is a divisor of integer f
                  if and only there is an integer g such that f=g*h

                  g=137 is a divisor of
                  f=137 because there is an integer, namely
                  h=1, such that f=g*h

                  g=137 is a divisor of
                  f=0 because there is an integer, namely
                  h=0, such that f=g*h

                  g=137 is the largest integer that is a divisor of
                  both 0 and 137, since it the largest divisor of 137.

                  hence gcd(0,137)=137

                  Some folk might not like that,
                  but any alternative is sure to be far worse,
                  structurally.

                  Notice that no-where have we divided by zero.

                  d
                • Michael Bell
                  0/0 shouldn t be defined as zero, consider the following: What is x/x as x-- 0? What is (x*x)/x as x-- 0? What is x/(x*x) as x-- 0? This clearly shows that
                  Message 8 of 11 , Jan 8, 2002
                    0/0 shouldn't be defined as zero, consider the following:
                    What is x/x as x-->0? What is (x*x)/x as x-->0? What is x/(x*x) as x-->0?
                    This clearly shows that 0/0 can take the values 0, 1 and infinity, and
                    indeed cna be made to tak any value you care to name. So "undefined" is
                    definitely the best thing for it.

                    Michael.

                    > I think 0/0 should be defined as 0. 0 multiplied by any non-zero number
                    > gives 0, and I don't see why multiplying it by infinity should be any
                    > different.
                    >
                    > If this is not so, then I could take my zero Euro's, and divide them into
                    my
                    > zero people, and end up extremely wealthy.
                    >
                    > Jon Perry
                  • Jon Perry
                    Reminds me of the riddle: What is apple multiplied by car? Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
                    Message 9 of 11 , Jan 9, 2002
                      Reminds me of the riddle:

                      What is apple multiplied by car?

                      Jon Perry
                      perry@...
                      http://www.users.globalnet.co.uk/~perry
                      http://www.users.globalnet.co.uk/~perry/maths
                      BrainBench MVP for HTML and JavaScript
                      http://www.brainbench.com


                      -----Original Message-----
                      From: Michael Bell [mailto:mike.d.bell@...]
                      Sent: 08 January 2002 20:24
                      To: Primes List
                      Subject: Re: [PrimeNumbers] New primers


                      0/0 shouldn't be defined as zero, consider the following:
                      What is x/x as x-->0? What is (x*x)/x as x-->0? What is x/(x*x) as x-->0?
                      This clearly shows that 0/0 can take the values 0, 1 and infinity, and
                      indeed cna be made to tak any value you care to name. So "undefined" is
                      definitely the best thing for it.

                      Michael.

                      > I think 0/0 should be defined as 0. 0 multiplied by any non-zero number
                      > gives 0, and I don't see why multiplying it by infinity should be any
                      > different.
                      >
                      > If this is not so, then I could take my zero Euro's, and divide them into
                      my
                      > zero people, and end up extremely wealthy.
                      >
                      > Jon Perry




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