[ExI] Goldbach Conjecture

Will Steinberg asyluman at gmail.com
Sun Nov 29 17:16:10 UTC 2009

Here's a little bit of LaTeX delineating some stuff.  Why the end works is
that the overflow from P/p (to get the fractional part) will always be an
even number (prime-prime) and the fractional part from 2n/p will be 2n-p
which will always be odd, unless p is 2, in which case the first part is odd
and the second even.  In any case, the fparts are not equal and so there is
a column in the matrix for ANY 2n satisfying Goldbach.  QED.

On Sat, Nov 28, 2009 at 6:39 PM, Will Steinberg <asyluman at gmail.com> wrote:

> I think I've got it now.  P != 2nmodp +pk, right?  We can re-express 2nmodp
> as 2n-p*[2n/p], where brackets denote the floor function.  So we have P !=
> 2n - p*[2n/p] +pk.  Dividing by p yields the following:
> P/p != 2n/p - [2n/p] +k
> 2n/p-[2n/p] equals the fractional part of 2n/p.  And since k is any
> integer, we can set it equal to the integer part of P/p.  Now we have this:
> frac(P/p) != frac(2n/p).
> Now: since P and p are prime, the fractional part will be irreducible.  And
> since p is in the denominator of 2n/p, it too must have an irreducible
> fractional part.  But since 2 is in the numerator, these parts must be
> different!  The only case in which they would not be is if n=p, in which
> case 2n would be a double of a prime, satisfying the GC.  I think this is
> really it.
> On Sat, Nov 28, 2009 at 11:55 AM, Will Steinberg <asyluman at gmail.com>wrote:
>> If it can be proved that every two-way sieve of eratosthenes has at least
>> one hole, the conjecture can be proven.  What this means is that (since oles
>> are at 2k, 3k, 5k, nmod2+2k, nmod5+5k, etc.)  There has got to be some kind
>> of proof saying that for any given number n, there is a prime in n than
>> cannot be expressed by nmodp+pk.  What has to be looked at is the modulo
>> values that will be given for ns. I think we can often choose 3, because the
>> only case when 2 can be covered is if we have nmod2=1 and pk=2.  (or if the
>> number is divisible by 3).  Otherwise we can simply continue moving up our
>> primes.  In the case of 2n=22, we see holes at 3,5, and 11.  What is needed
>> to continue is a way to prove there will always be a p that doesn't equal
>> nmodp +pk
>> 2009/11/28 spike <spike66 at att.net>
>>> > -----Original Message-----
>>> > From: extropy-chat-bounces at lists.extropy.org
>>> > [mailto:extropy-chat-bounces at lists.extropy.org<extropy-chat-bounces at lists.extropy.org>]
>>> On Behalf Of
>>> > Giulio Prisco (2nd email)
>>> > Sent: Friday, November 27, 2009 11:12 PM
>>> > To: ExI chat list
>>> > Subject: Re: [ExI] Goldbach Conjecture
>>> >
>>> > I think the Goldbach conjecture is probably false, with
>>> > probability 1 (that means, certainly false). Here is why:
>>> >
>>> > Apparently there is nothing in the laws of arithmetics that
>>> > forces an even number to be the sum of two prime numbers. The
>>> > conjecture is true for all even numbers on which it has been
>>> > tested, but these are an infinitesimal part of the total (any
>>> > finite number is infinitesimal wrt infinite). Hence, if there
>>> > is no proof, the probability of he Goldbach conjecture being
>>> > true is zero.
>>> I disagree sir, however I confess my line of reasoning is not as well
>>> developed as the one you offer.
>>> I took the even numbers and calculated the number of ways each even
>>> number (shown on the X axis) could be expressed as a the sum of two primes.
>>> The number of different ways is on the Y.  For Goldbach to have been wrong,
>>> there is some super-anomaly way out there somewhere which departs from the
>>> data trends shown.
>>> Yes I do know that this line of reasoning is not to be substituted for
>>> actual mathematical logic, do forgive please.
>>> I plotted them to a few million on matlab, found there are striking
>>> patterns in the data, such as the eye-catching streaks.
>>> spike
>>> _______________________________________________
>>> extropy-chat mailing list
>>> extropy-chat at lists.extropy.org
>>> http://lists.extropy.org/mailman/listinfo.cgi/extropy-chat
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