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

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Ben Bacarisse

Posts: 1,972
Registered: 7/4/07
Re: Vindication of Goldbach's conjecture
Posted: Jul 21, 2012 10:14 PM
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mluttgens <> writes:

> On 21 juil, 15:32, Ben Bacarisse <> wrote:
>> 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?)

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

Thanks. Much better.

> 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

> 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

> 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

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