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Re: gcd conjecture
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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 0 Attachment
 In primenumbers@y..., "djbroadhurst" <d.broadhurst@o...> wrote:> Yes, Marcel, that is something that I noted
So if there were a contest for the best proof of a false statement,
> about both you and Jack: you seem incapable of
> proving false statments while several other
> contributors claim to transcend this limitation:)
I guess I would not be expected to do well. :) 0 Attachment
> So if there were a contest for the best proof of a false statement,
\begin{silly}
> I guess I would not be expected to do well. :)
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 0 Attachment
 Paul Leyland <pleyland@...> wrote:> > So if there were a contest for the best proof of a false
Only two? Hmmm, maybe I've got my pedantic hat on tonight, as I
> 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.
needed more than two fingers to count the errors. ;)
However, I'm convinced, all horses are white.
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
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Some false proofs can be valuable:
http://wwwgap.dcs.stand.ac.uk/~history/Mathematicians/Kempe.html 0 Attachment
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 wellordered.) Fix a wellordering 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''), wellorder
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 wellknown
(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,
\begin{silly}
> I guess I would not be expected to do well. :)
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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> 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
Pedantic, or just a careless reader? I wrote AT LEAST two flaws,
> needed more than two fingers to count the errors. ;)
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 0 Attachment
 Paul Leyland <pleyland@...> wrote:> > From: Phil Carmody [mailto:thefatphil@...]
A careless reader. I must stop spinning my wheels.
> >  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
Flaw 4 could be seen as an element in the group closure formed by the
> deduce
> that you found at least four flaws and possibly at least 2^n where
> n is
> a small integer.
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
So I assume you use hand orientation for a sign bit?
> fingers
> of one hand are prime, whether I use unary or binary
> representation.
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
__________________________________________________
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 Paul Leyland <pleyland@...> wrote:> > From: Phil Carmody [mailto:thefatphil@...]
A careless reader. I must stop spinning my wheels.
> >  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
Flaw 4 could be seen as an element in the group closure formed by the
> deduce
> that you found at least four flaws and possibly at least 2^n where
> n is
> a small integer.
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
So I assume you use hand orientation for a sign bit?
> fingers
> of one hand are prime, whether I use unary or binary
> representation.
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 0 Attachment
 djbroadhurst <d.broadhurst@...> wrote:> Some false proofs can be valuable:
Didn't Gabriel Lam� accidentally invent nonUFDs in his failed
> http://wwwgap.dcs.stand.ac.uk/~history/Mathematicians/Kempe.html
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
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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
ObPedant:
>of one hand are prime, whether I use unary or binary representation.
>
>Paul
I assume you mean digits, using fingers alone neither number is prime.
Nathan 0 Attachment
> At 01:53 AM 6/26/2002 0700, Paul Leyland wrote:
Chambers 20th Century Dictionary, 1973 edition:
> >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
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 Chambers73 is the
only one I have at my fingertips.
Don't try to outpedant me. ;)
Paul
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