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Re: gcd conjecture

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  • djbroadhurst
    ... Yes, Marcel, that is something that I noted about both you and Jack: you seem incapable of proving false statments while several other contributors claim
    Message 1 of 12 , Jun 24, 2002
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      Marcel Martin wrote:
      > It's a good thing I failed to prove it true :-)
      Yes, Marcel, that is something that I noted
      about both you and Jack: you seem incapable of
      proving false statments while several other
      contributors claim to transcend this limitation:-)
      David
    • jbrennen
      ... So if there were a contest for the best proof of a false statement, I guess I would not be expected to do well. :)
      Message 2 of 12 , Jun 25, 2002
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        --- In primenumbers@y..., "djbroadhurst" <d.broadhurst@o...> wrote:
        > Yes, Marcel, that is something that I noted
        > about both you and Jack: you seem incapable of
        > proving false statments while several other
        > contributors claim to transcend this limitation:-)

        So if there were a contest for the best proof of a false statement,
        I guess I would not be expected to do well. :)
      • Paul Leyland
        ... begin{silly} Theorem: All primes are odd. Proof: All even numbers are divisible by 2. Therefore all numbers other than 2 are composite. This leaves
        Message 3 of 12 , Jun 25, 2002
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          > So if there were a contest for the best proof of a false statement,
          > I guess I would not be expected to do well. :)

          \begin{silly}

          Theorem: All primes are odd.

          Proof: All even numbers are divisible by 2. Therefore all numbers
          other than 2 are composite. This leaves only the case of 2 to consider.
          If it is prime, it is certainly a very odd case. We have a
          contradiction, so all primes are odd.

          \end{silly}


          There are at least two flaws in the above proof. One ought to be
          obvious. I suspect a number of readers (not including Jack, David,
          Marcel and Phil) will have difficulty finding even this one.


          Paul
        • Phil Carmody
          ... Only two? Hmmm, maybe I ve got my pedantic hat on tonight, as I needed more than two fingers to count the errors. ;-) However, I m convinced, all horses
          Message 4 of 12 , Jun 25, 2002
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            --- Paul Leyland <pleyland@...> wrote:
            > > So if there were a contest for the best proof of a false
            > statement,
            > > I guess I would not be expected to do well. :)
            >
            > \begin{silly}
            >
            > Theorem: All primes are odd.
            >
            > Proof: All even numbers are divisible by 2. Therefore all numbers
            > other than 2 are composite. This leaves only the case of 2 to
            > consider.
            > If it is prime, it is certainly a very odd case. We have a
            > contradiction, so all primes are odd.
            >
            > \end{silly}
            >
            >
            > There are at least two flaws in the above proof. One ought to be
            > obvious. I suspect a number of readers (not including Jack, David,
            > Marcel and Phil) will have difficulty finding even this one.

            Only two? Hmmm, maybe I've got my pedantic hat on tonight, as I
            needed more than two fingers to count the errors. ;-)

            However, I'm convinced, all horses are white.

            Phil

            =====
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            losing the ability to manage files on the user's own
            hard disk." - Prof. Stuart E Madnick, MIT.
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          • djbroadhurst
            Some false proofs can be valuable: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Kempe.html
            Message 5 of 12 , Jun 25, 2002
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            • Insall
              Theorem: All integers are interesting. Proof: Let S be the set of interesting integers, and let X be its complement. Suppose that X is nonvoid, and let M
              Message 6 of 12 , Jun 25, 2002
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                Theorem: All integers are interesting.

                Proof:
                Let S be the set of interesting integers, and let X be its complement.
                Suppose that X is nonvoid, and let M denote the set of all absolute values
                of members of X. Then M is a nonempty set of nonnegative integers, and so
                it has a least member, m. Now, m is the least absolute value of a
                noninteresting integer, so m is interesting. Let p be a member of X that
                has absolute value equal to m. If p=m, then p is interesting, i.e. p is in
                S. But since p is in X, p is not in S, so p is not equal to m. Since m is
                the absolute value of p, it must be the case that p=-m. Thus p is the
                negative of the least absolute value of a noninteresting integer, so p is
                interesting. This is a contradiction, so the set M is a set of nonnegative
                integers with no least member, a contradiction. Hence all integers are
                interesting.
                qed

                Theorem: All rational numbers are interesting.

                Proof:
                Enumerate all the rational numbers: Q={r_k | k is a natural number}.
                Suppose there are some uninteresting rational numbers, and let U be the set
                of all indices k such that r_k is uninteresting. The set U has a least
                member, say m. Then r_m is the first rational number listed that is
                uninteresting. Isn't that interesting? Yes. Thus, r_m is an interesting
                rational number, contrary to hypothesis,so the theorem follows.
                qed

                Theorem [WO(R)]: Every real number is interesting.

                Proof:
                (Note: The notation ``WO(R)'' means we assume for the proof that the
                real numbers can be well-ordered.) Fix a well-ordering of the real numbers,
                and let U denote the set of uninteresting reals. Let m be the first member
                of the set U. Then m is the first uninteresting real number. This is so
                interesting I can hardly contain my exuberation, so m is necessarily an
                interesting real number. This contradiction proves the theorem.
                qed


                Theorem [V=L]: Every set is interesting.

                Proof:
                Using the constructibility assumption (denoted by ``V=L''), well-order
                the universe of all sets. If there were an uninteresting set, then there
                would be a first uninteresting set. It has been known since Goedel that
                this would be an extremely interesting set of circumstances, so the first
                uninteresting set is interesting. It follows that every (constructible) set
                is interesting.
                qed

                Corollary: Every set is interesting.

                Proof:
                By the above, all constructible sets are interesting. It is well-known
                (cf [anybody, anytime]) that any set that is not constructible must be very
                interesting indeed. Hence all sets are interesting, as claimed.
                qed


                Main Theorem: Everything in mathematics is interesting.

                Proof:
                It has been claimed that every mathematical concept can be represented
                in terms of sets. Since every set is interesting, every such concept is
                interesting, being represented by an interesting set. Since this is only a
                claim, and there are some detractors, consider any specific mathematical
                concept that is not representable in the theory of sets. Such a concept
                must be stupendously interesting, so of course, everything in mathematics is
                interesting, as the theorem purports.
                qed

                -----Original Message-----
                From: Paul Leyland [mailto:pleyland@...]
                Sent: Tuesday, June 25, 2002 3:59 PM
                To: jbrennen; primenumbers@yahoogroups.com
                Subject: RE: [PrimeNumbers] Re: proving false statements



                > So if there were a contest for the best proof of a false statement,
                > I guess I would not be expected to do well. :)

                \begin{silly}

                Theorem: All primes are odd.

                Proof: All even numbers are divisible by 2. Therefore all numbers
                other than 2 are composite. This leaves only the case of 2 to consider.
                If it is prime, it is certainly a very odd case. We have a
                contradiction, so all primes are odd.

                \end{silly}


                There are at least two flaws in the above proof. One ought to be
                obvious. I suspect a number of readers (not including Jack, David,
                Marcel and Phil) will have difficulty finding even this one.


                Paul


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              • Paul Leyland
                ... Pedantic, or just a careless reader? I wrote AT LEAST two flaws, leaving open the possibility there may be more than two. Anyway, from your comment about
                Message 7 of 12 , Jun 26, 2002
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                  > From: Phil Carmody [mailto:thefatphil@...]
                  > --- Paul Leyland <pleyland@...> wrote:
                  > > \begin{silly}
                  > >
                  > > Theorem: All primes are odd.
                  > >
                  > > Proof: All even numbers are divisible by 2. Therefore all numbers
                  > > other than 2 are composite. This leaves only the case of 2 to
                  > > consider.
                  > > If it is prime, it is certainly a very odd case. We have a
                  > > contradiction, so all primes are odd.
                  > >
                  > > \end{silly}
                  > >
                  > >
                  > > There are at least two flaws in the above proof. One ought to be
                  ...
                  > Only two? Hmmm, maybe I've got my pedantic hat on tonight, as I
                  > needed more than two fingers to count the errors. ;-)

                  Pedantic, or just a careless reader? I wrote AT LEAST two flaws,
                  leaving open the possibility there may be more than two.

                  Anyway, from your comment about needing more than two fingers I deduce
                  that you found at least four flaws and possibly at least 2^n where n is
                  a small integer.

                  Sorry. Feeling grumpy this morning.

                  ObPrime: the largest integers to which I can easily count on the fingers
                  of one hand are prime, whether I use unary or binary representation.

                  Paul
                • Phil Carmody
                  ... A careless reader. I must stop spinning my wheels. ... Flaw 4 could be seen as an element in the group closure formed by the first three. Somewhat like
                  Message 8 of 12 , Jun 26, 2002
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                    --- Paul Leyland <pleyland@...> wrote:
                    > > From: Phil Carmody [mailto:thefatphil@...]
                    > > --- Paul Leyland <pleyland@...> wrote:
                    > > > \begin{silly}
                    > > >
                    > > > Theorem: All primes are odd.
                    > > >
                    > > > Proof: All even numbers are divisible by 2. Therefore all
                    > numbers
                    > > > other than 2 are composite. This leaves only the case of 2 to
                    > > > consider.
                    > > > If it is prime, it is certainly a very odd case. We have a
                    > > > contradiction, so all primes are odd.
                    > > >
                    > > > \end{silly}
                    > > >
                    > > >
                    > > > There are at least two flaws in the above proof. One ought to
                    > be
                    > ...
                    > > Only two? Hmmm, maybe I've got my pedantic hat on tonight, as I
                    > > needed more than two fingers to count the errors. ;-)
                    >
                    > Pedantic, or just a careless reader? I wrote AT LEAST two flaws,
                    > leaving open the possibility there may be more than two.

                    A careless reader. I must stop spinning my wheels.

                    > Anyway, from your comment about needing more than two fingers I
                    > deduce
                    > that you found at least four flaws and possibly at least 2^n where
                    > n is
                    > a small integer.

                    Flaw 4 could be seen as an element in the group closure formed by the
                    first three. Somewhat like pairs in cribbage.

                    > Sorry. Feeling grumpy this morning.

                    You can cheer yourself up by contributing positively to the thread
                    beginning with my next post, which will display my great confusion in
                    matters that are probably a breeze for you.

                    > ObPrime: the largest integers to which I can easily count on the
                    > fingers
                    > of one hand are prime, whether I use unary or binary
                    > representation.

                    So I assume you use hand orientation for a sign bit?
                    And I also assume you don't count using a Gray code?
                    :-)

                    ObBadJoke: Why did the 4 computer science students get kicked out of
                    their noisy college bar?
                    Because they tried to order a round of drinks using just hand
                    signals.

                    It doesn't warrant explanation, but if you don't get it note that
                    also 'fails' for 12 students.

                    Phil


                    =====
                    --
                    "One cannot delete the Web browser from KDE without
                    losing the ability to manage files on the user's own
                    hard disk." - Prof. Stuart E Madnick, MIT.
                    So called "expert" witness for Microsoft. 2002/05/02

                    __________________________________________________
                    Do You Yahoo!?
                    Yahoo! - Official partner of 2002 FIFA World Cup
                    http://fifaworldcup.yahoo.com
                  • Phil Carmody
                    ... A careless reader. I must stop spinning my wheels. ... Flaw 4 could be seen as an element in the group closure formed by the first three. Somewhat like
                    Message 9 of 12 , Jun 26, 2002
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                      --- Paul Leyland <pleyland@...> wrote:
                      > > From: Phil Carmody [mailto:thefatphil@...]
                      > > --- Paul Leyland <pleyland@...> wrote:
                      > > > \begin{silly}
                      > > >
                      > > > Theorem: All primes are odd.
                      > > >
                      > > > Proof: All even numbers are divisible by 2. Therefore all
                      > numbers
                      > > > other than 2 are composite. This leaves only the case of 2 to
                      > > > consider.
                      > > > If it is prime, it is certainly a very odd case. We have a
                      > > > contradiction, so all primes are odd.
                      > > >
                      > > > \end{silly}
                      > > >
                      > > >
                      > > > There are at least two flaws in the above proof. One ought to
                      > be
                      > ...
                      > > Only two? Hmmm, maybe I've got my pedantic hat on tonight, as I
                      > > needed more than two fingers to count the errors. ;-)
                      >
                      > Pedantic, or just a careless reader? I wrote AT LEAST two flaws,
                      > leaving open the possibility there may be more than two.

                      A careless reader. I must stop spinning my wheels.

                      > Anyway, from your comment about needing more than two fingers I
                      > deduce
                      > that you found at least four flaws and possibly at least 2^n where
                      > n is
                      > a small integer.

                      Flaw 4 could be seen as an element in the group closure formed by the
                      first three. Somewhat like pairs in cribbage.

                      > Sorry. Feeling grumpy this morning.

                      You can cheer yourself up by contributing positively to the thread
                      beginning with my next post, which will display my great confusion in
                      matters that are probably a breeze for you.

                      > ObPrime: the largest integers to which I can easily count on the
                      > fingers
                      > of one hand are prime, whether I use unary or binary
                      > representation.

                      So I assume you use hand orientation for a sign bit?
                      And I also assume you don't count using a Gray code?
                      :-)

                      ObBadJoke: Why did the 4 computer science students get kicked out of
                      their noisy college bar?
                      Because they tried to order a round of drinks using just hand
                      signals.

                      It doesn't warrant explanation, but if you don't get it note that
                      also 'fails' for 12 students.

                      Phil


                      =====
                      --
                      "One cannot delete the Web browser from KDE without
                      losing the ability to manage files on the user's own
                      hard disk." - Prof. Stuart E Madnick, MIT.
                      So called "expert" witness for Microsoft. 2002/05/02

                      __________________________________________________
                      Do You Yahoo!?
                      Yahoo! - Official partner of 2002 FIFA World Cup
                      http://fifaworldcup.yahoo.com
                    • Phil Carmody
                      ... Didn t Gabriel Lamé accidentally invent non-UFDs in his failed attempt (i.e. false proof) to generalise his n=7 FLT proof to arbitrary numbers? Phil =====
                      Message 10 of 12 , Jun 26, 2002
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                        --- djbroadhurst <d.broadhurst@...> wrote:
                        > Some false proofs can be valuable:
                        > http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Kempe.html

                        Didn't Gabriel Lam� accidentally invent non-UFDs in his failed
                        attempt (i.e. false proof) to generalise his n=7 FLT proof to
                        arbitrary numbers?

                        Phil



                        =====
                        --
                        "One cannot delete the Web browser from KDE without
                        losing the ability to manage files on the user's own
                        hard disk." - Prof. Stuart E Madnick, MIT.
                        So called "expert" witness for Microsoft. 2002/05/02

                        __________________________________________________
                        Do You Yahoo!?
                        Yahoo! - Official partner of 2002 FIFA World Cup
                        http://fifaworldcup.yahoo.com
                      • Nathan Russell
                        ... ObPedant: I assume you mean digits, using fingers alone neither number is prime. Nathan
                        Message 11 of 12 , Jul 1, 2002
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                          At 01:53 AM 6/26/2002 -0700, Paul Leyland wrote:
                          >ObPrime: the largest integers to which I can easily count on the fingers
                          >of one hand are prime, whether I use unary or binary representation.
                          >
                          >Paul

                          ObPedant:

                          I assume you mean digits, using fingers alone neither number is prime.

                          Nathan
                        • Paul Leyland
                          ... Chambers 20th Century Dictionary, 1973 edition: finger, n. one of the five terminal parts of the hand, ... There are other definitions, but that s the
                          Message 12 of 12 , Jul 2, 2002
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                            > At 01:53 AM 6/26/2002 -0700, Paul Leyland wrote:
                            > >ObPrime: the largest integers to which I can easily count on
                            > the fingers
                            > >of one hand are prime, whether I use unary or binary representation.
                            > >
                            > >Paul
                            >
                            > ObPedant:
                            >
                            > I assume you mean digits, using fingers alone neither number is prime.
                            >
                            > Nathan


                            Chambers 20th Century Dictionary, 1973 edition:

                            finger, n. one of the five terminal parts of the hand, ...


                            There are other definitions, but that's the principal one. Other
                            editions very likely have the same definition but Chambers-73 is the
                            only one I have at my fingertips.

                            Don't try to out-pedant me. ;-)


                            Paul
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