Date: Jul 24, 2012 9:20 AM Author: mluttgens Subject: Re: Vindication of Goldbach's conjecture On 22 juil, 04:14, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:

> 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.

Till now, I did not find a clue in the litterature, but you have

perhaps a reference?

Marcel Luttgens