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Re: A Proof of Goldbach's Conjecture
Posted:
May 8, 2012 3:36 PM
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On Friday, May 4, 2012 9:50:18 PM UTC+1, quasi wrote:
> "dcowham@yahoo.co.uk" wrote:
> >quasi wrote:
> >> "dcowham@yahoo.co.uk" wrote:
> >>
> >> >Well I thought I was getting somewhere with Quasi until
> >> >he cut me dead in my tracks.
> >>
> >> For good reasons.
> >>
> >> You were asked to provide a road map to your proof, but from
> >> my perspective, your attempt at that did nothing to clarify
> >> its structure.
> >>
> >> It now seems likely that
> >>
> >> (1) You're actually incapable of expressing your arguments
> >> coherently.
> >>
> >> (2) You may not actually be able to understand the flaws in
> >> your proof (flaws which are surely there) even if they are
> >> explained to you.
> >>
> >> I don't have much time these days, so getting involved in
> >> the mess of argument which you call a proof is a trap which,
> >> if I has any common sense, I should surely avoid.
> >>
> >> Still, since I like teaching, I'll dabble a bit, taking as a
> >> challenge the task of identifying the key flaws in your
> >> proof and trying to get you to understand the issues.
> >>
> >> If the communication gap is too big, I may back out, but in the
> >> meantime, let's see what happens.
> >>
> >> My gut feeling is that the proof fails (in an irreparable way)
> >> because:
> >>
> >> (1) The linear map from U' to V' which you claim is surjective
> >> is, in fact, not necessarily surjective. Moreover, I suspect
> >> that for N sufficiently large, the map is _never_ surjective.
> >>
> >> (2) Even if the map from U' to V' is surjective, that doesn't
> >> force the specified basis elements of U' to map to the
> >> specified basis elements of V', and without that, you don't
> >> get to conclude log(2N - p_i) = log(q_j). In other words, the
> >> map T sends the basis element log(p_i) of U' to the element
> >> log(2N - p_i) of V', but then you appear to assert that the
> >> element log(2N - p_i) of V' must, in fact, be equal to one of
> >> the basis elements log(q_j) of V'. As far as I can see, your
> >> justification for this last claim uses the following fatally
> >> flawed reasoning ...
> >>
> >> Firstly you assume that T restricted to U' is a surjection of
> >> U' onto V', which as I've already indicated, is a dubious claim.
> >>
> >> But let's suppose that claim is actually OK.
> >>
> >> So then consider a basis element log(q_j) of V'.
> >>
> >> By assumption, T restricted to U' is a surjection onto V',
> >> so some element of U' must map to log(q_i). Fine, no problem
> >> with that. But then you claim that since the linear map T
> >> restricted to U' is a surjection onto V', the element of U'
> >> which maps onto log(q_j) must have the form log(p_i), however
> >> that logic is fatally flawed. Of course, if you can get away
> >> with that, then
> >>
> >> T(log(p_i)) = log(q_j)
> >>
> >> => log(2N - p_i) = log(q_j)
> >>
> >> => 2N - p_i = q_j
> >>
> >> => p_i + q_j = 2N
> >>
> >> which shows that 2N is a sum of two primes.
> >>
> >> But the flaw is there and there's no getting around it.
> >>
> >> Since only one flaw is needed to defeat the proof, I'd rather
> >> not focus first on the issue of whether or not T restricted
> >> to U' must be a surjection of U' onto V'. Go ahead and assume
> >> it _is_ a surjection. Now show me how you get from that to the
> >> result T(log(p_i)) = log(q_j).
> >
> >Thanks for your reply! I don't think you quite understand my
> >arguments yet.
>
> Perhaps not, but at the same time, I don't think you made any
> attempt to follow the points of my above critique.
>
> >T has to map U' onto V' due to the way the algorithm operates
> >and I believe this must be a surjection if it doesn't calculate
> >any instances of Goldbach's Conjecture before terminating.
>
> That's just an assertion, and until justified further with more
> detail and greater clarity than in your proof attempt,I don't
> believe the assertion is valid. More precisely, I accept that
> T restricted to U' is a linear map from U' to V', but I don't
> accept your claim that it must be a surjection.
>
> Still, I said I was willing to temporarily accept that
> assertion so as to focus on the issue of how that leads to your
> claim that log(2N - p_i) = log(q_j) for some primes p_i,q_j.
>
> >However I am also convinced that in this scenario U' could
> >not attain it's maximum theoretical size and so at least one
> >other log(p_i) of U (not in U') must form a scenario in
> >which log(2N - p_i) = log(q_j).
>
> Another assertion without any justification other than the
> argument in your unreadable proof attempt.
>
> >Does that clarify things?
>
> No, not in the least.
>
> I objected to various claims in your proof and in response,
> you've simply repeated those claims.
>
> Your response shows zero effort your part.
>
> If you keep that up, this conversation won't last long.
>
> I gave reasons why the surjectivity of T' (temporarily
> granting the claim that the linear map T' : U' - V' is a
> surjection) doesn't suffice to yield the desired conclusion
> log(2N - p_i) = log(q_j).
>
> You need to try harder to understand the points of my
> objection (perhaps in a general context unrelated to your
> proof) before you have any chance of deflecting it.
>
> quasi
Hi Quasi,
Thanks for your reply although I'm sorry you still don't really understand my proof and assertions which I will hope to address now!
T has to map U' onto V' by definition when it hasn't computed any instances of Goldbach's Conjecture and, as stated, I believe this must form a surjective linear map. I believe that in this scenario the only reason that no instances could be computed in (2, [2N/3]) is that all rows corresponding to these instances of GC, when we consider our entire linear map as a whole, have to be entirely zero. If that is so then it would be impossible to compute any instances of GC and we can say with certainty that U' could never attain it's maximum possible size in that scenario. I spend a lot of time in my proof trying to explain why GC must still hold true in that particular scenario ie trying to prove why our instances of GC must coincide with these zero rows. The only case I have found to verify this is 2N=128 and in that particular example U' could never contain the basis elements log(19) and log(31).
I hope that clarifies things?
Regards,
David
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