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Topic: Vindication of Goldbach's conjecture
Replies: 5   Last Post: Jul 25, 2012 9:51 AM

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Gus Gassmann

Posts: 28
Registered: 9/11/09
Re: Vindication of Goldbach's conjecture
Posted: Jul 24, 2012 11:01 AM
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On Jul 24, 10:20 am, mluttgens <lutt...@gmail.com> wrote:
> On 22 juil, 04:14, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
>
>
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>
>
>

> > mluttgens <lutt...@gmail.com> writes:
> > > On 21 juil, 15:32, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
> > >> lutt...@gmail.com writes:
>
> > >> <snip>
>
> > >> > Both terms of 6 = 3 + 3 are primes.
> > >> > I considered the case where at least one of the terms is not prime.
> > >> > Your example is irrelevant!

>
> > >> Your claim is essentially the same as GC.  I thought you'd miss-worded
> > >> it which is why I thought there was a counter example.  Correctly worded
> > >> (as I think it is) proving it is equivalent to proving GC.

>
> > >> <snip>
>
> > >> >> ... counter examples may
> > >> >> be very hard to find, but that does not constitute a sound argument: you
> > >> >> can't prove X by noting that X follows from Y and challenging people to
> > >> >> disprove Y (but you know that, yes?).

>
> > >> > No, you did not.
>
> > >> What does that mean?  Is it a comment on my remark about your "proof by
> > >> you can't contradict me" method?

>
> > > Not at all.I was referring to some quibbling of you...
>
> > Ah, then better to put it next to the quibble.  In my opinion, the
> > comment about you proof structure ("look, you always get two primes if
> > you add and subtract some even number -- show me a counter example") was
> > much more than a quibble.

>
> > >> (By the way, can you get you newsreader to stop turning plain 7-bit
> > >> characters into HTML entities?)

> > <snip>
> > > Sorry, the new Goggle interface was responsible. For that reason,
> > > I have just went back to the older interface.

>
> > Thanks.  Much better.
>
> > <snip>
>
> > > Proof of the validity of Goldbach's conjecture
> > > _______________________________________

>
> > > According to the conjecture, every even integer greater than 4 can be
> > > expressed as the sum of two primes.

>
> > > Let?s consider the infinite series of uneven integers.
> > > Such series contains an infinite number of products p = ab, where a
> > > and b are primes.
> > > To each product p corresponds a single sum s = a + b, s being of
> > > course an even integer.
> > > This approach leads to all possible sums of two primes.

>
> > There's no point to this pre-amble.  It adds nothing to the discussion
> > and just looks like padding.

>
> > > By the way, some even integers can be the sum of two uneven integers,
> > > at least one of them not being a prime.

>
> > All even integers other than zero can be written as the sum of two odd
> > integers, at least one of them not being prime: 2k = 1 + (2k-1).  It
> > comes over as a bit odd to say "some" when you are stating an obvious
> > property of all numbers != 0.

>
> > > This leads to the bold assumption, that one or more even numbers
> > > greater than 4 could not necessarily be expressed as the sum of 2
> > > primes.

>
> > I'd start the argument here...  You don't need (or use) any of the
> > above.

>
> > > A sum s of two primes a and b greater than 3 can always be written as
> > > s = (a + n) + (b - n) or s = (a ? n) + (b + n), where n is an even
> > > integer.
> > > The obtained terms (a +/- n) and/or (b -/+ n) can be prime numbers,
> > > but being ordinary uneven numbers does not imply that an even integer
> > > cannot be a sum of two primes.
> > > Let?s notice that such method, which consists of adding or
> > > subtracting  the successive elements of  the series of even numbers n,
> > > can be applied for arbitrarily large sums s.
> > > It leads to all possible pairs of numbers: two primes, a prime and a
> > > uneven number, that is  not  a prime, or two uneven numbers, which are
> > > not prime.

>
> > ...and you are assured of getting two primes for all s, only if GC is
> > true.

>
> > > On the other hand, a sum s? of two uneven integers, where at least one
> > > of its terms is not prime, can be transformed into a sum s of primes
> > > by adding some even integer n to one of its terms and subtracting the
> > > same n from  its other term.

>
> > This statement needs a proof.  If GC is true is it's obviously true; if
> > GC is false, it's false.

>
> > > To determine n, it suffices to apply the above method to the sum s =
> > > s?. Then, one straightforwardly gets the value of n leading to the
> > > uneven terms of sum s?.

>
> > The above is not a method of getting two primes -- it's a method of
> > getting all pairs of odd numbers that sum to s.  One of these will
> > always be a pair of primes only if GC is true.

>
> > > Example:
>
> > > s? = 13 + 15 = 28 (s? is not the sum of two primes).
>
> > > From s = s? = 28, one gets
> > > s = 5+23 and 11+17, and also
> > > s = (5+8) + (23-8) = 13+15 = s?
> > > s = (11+2) + (17-2) = 13+15 = s?
> > > QED!

>
> > > The assumption that one or more even numbers greater than 4 could not
> > > be expressed as the sum of 2 primes is thus refuted.

>
> > > This leads to the conclusion that any even integer can indeed be
> > > expressed as the sum of two primes.

>
> > > Marcel Luttgens
>
> > > July 22, 2012
>
> > No need to date your posts.  Usenet records the date of posting in the
> > header.

>
> > --
> > Ben.

>
> Thank you! You are of course right.
>
> But my aim was to show that a sum s? = a + b of two uneven numbers, at
> least one of them not being a prime, could easily be transformed into
> a sum of two primes, simply by adding and subtracting some even number
> from its terms:
>
> The chosen example was:
>
> s? = 13 + 15 =  (13-8) + (15+8) = 5+23
>                     =  (13-2) + (15+2) = 11+17
>
> It has been claimed that such transformation could sometimes not be
> possible.
> I am wondering about which terms a and b should be chosen to justify
> that claim.


That is, of course, a good question. IF Goldbach's conjecture is
false, then of course there is a counterexample to your claim.
However, nobody knows at this point one way or the other. However,
asking for a counterexample and not receiving one is by no means
equivalent to having found a proof! In essence your approach is this:

Theorem: Goldbach's conjecture is true.

Proof: If it were false, there would be a counterexample. Nobody has
found one. So the theorem is proven.

That is not mathematics, and I hope you can see why it isn't.




> Till now, I did not find a clue in the litterature, but you have
> perhaps a reference?
>
> Marcel Luttgens






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