Bill Dubuque wrote:
> malcobe@[EMAIL PROTECTED]
wrote:
> >
> > I keep on believing that the key idea [in Euclid's proof that there
> > are infinitely many primes] is the construction of the number
> > "the product of all primes plus one"
>
> Actually the "reason" such proofs work is because the ring of integers
> has relatively few units. Thus, by way of the Pigeonhole Principle,
Not true, because Bill is under the assumption Natural-Numbers =
Finite
Integers, which is an ill-defined set. Things are different in Natural-
Numbers
= Infinite Integers.
The reason that "multiply the lot add 1" works is that it is merely
another form
of primes such as 2k+1 or 2k-1. One form that does not work is 2k for
2 is the
only even prime number. Twin Primes are of the form k,k+2 such as 3,5
but there
are primes of form k,k+2,k+4 such as 3,5,7.
Now Bill with his old fake math of Natural-Numbers = Finite Integers
would
be unable to launch into the question of whether there is a infinitude
of primes
such as 3,5,7, but when you do mathematics with the true-blue counting
numbers
the Infinite Integers there is a simple constructive proof of the
infinitude of Triplet Primes.
An infinitude of Natural Numbers whose last digit is 9, 1, and 3 and
then those whose last digit
is 7, 9, and 1. Obviously no number whose last digit is 5 is prime. So
what Bill
thinks is the underpinning of these infinitude proofs with some
Pigeonhole Principle is far
off the mark. The underpinning is that almost any form for primes is
an infinite set, not all sets
but most.
>
> These are the sort of results that a curious student can easily
> discover independently when pondering generalizations of Euclid's
> theorem (indeed I found them as a teenager). They make excellent
So you are doing Ring theory as a teenager? But when reading Hardy's
book, unable to spot a mistake in his Euclid Infinitude of Primes
proof?
To me, a mathematics proof is like the electrical system wiring of an
entire
house. The lights and appliances work because everything is connected
and there are no shorts or grounds. But many proofs of mathematics
have so
many holes and gaps, that no electricity works for them. Many proofs
in mathematics
would be like a electrician coming to wire a house and throwing down
some wire
and then leaving.
> exercises for beginning number theory students. If I can dig up
> my original notes I will post further results. I hope readers
> will contribute some of their favorites (there are many other
> variants of proofs of Euclid's theorem listed in Ribenboim's
> Book of Prime Number Records, but not those I presented above).
>
> --Bill Dubuque
I challenge Bill to write a Euclid Infinitude of Primes proof both
direct and indirect. I challenge him
because I want to discuss Metamathematics
of the differences in the pattern of Direct versus Indirect. I cannot
discuss that if the person has no
reference to a mathematics proof that has a Direct and Indirect. Most
mathematicians believe that
if they provide a proof of a subject, that they can switch to a
indirect or direct, freely. So if they get hold
of one, they believe they can turn around and provide the other
method. I do not share that opinion, because
I know that in geometry proofs, the indirect method is virtually
nonexistent and that only one method of
proving occurrs-- some form of direct method.
The method of proofs-- direct or indirect, I feel comes more from
physics than it comes from mathematics
or logic and is meta-mathematics. This subject of an analysis of
direct versus indirect method is
seldom discussed and there is a paucity of anyone researching this
subject.
So the Euclid Infinitude of Primes is a pretty example of a proof that
has both Direct and Indirect and
allows for analysis of these two methods. I believe they are
Complimentarity methods, and that means
if a proof is of one, then it may or may not have the other.
Complimentarity is independence of one another.
So again, I challenge Bill, if not too scared, to write a Euclid IP in
direct and indirect. Then we can
analyze the Metamathematics of the two methods.
The way I see it, the direct method versus indirect method is like the
physics of electron versus
positron, where both have things in common -- same mass, but have
things opposite -- charge.
Both are independent of one another. And I hope to dispel this widely
perceived false notion
in the math community that once a proof is found, whether direct or
indirect, that they thence can
transform the proof into the other method.
Maybe Bill is too scared to do a Euclid IP both direct and indirect.
Maybe he is more of a politician and
dodging a challenge rather than simply offering his rendition. Maybe I
scare people, for fear of any
shortcoming.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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