From: cet1-nospam@cam.ac.uk.invalid (Chris Thompson)
Subject: Re: Q: For Golbach's Conjecture junkies
Date: 14 May 2000 15:41:47 GMT
Newsgroups: sci.math
Summary: [missing]

In article <1vrphs8on3bvd3t2htnh19cc75qe2ptqcj@4ax.com>,
Fred W. Helenius  <fredh@ix.netcom.com> wrote:
>"Mark Lamb" <mark.lamb@asu.edu> wrote:
>
>>  BTW - it has already been proven that
>>if there are exceptions to GC, there are a finite number of them (too lazy
>>to look up ref., but I may if someone really really insists :)
>
>I would very much like to see such a reference; the only "finite
>number of exceptions" results that I know are far weaker:
>
>Every odd number is the sum of three primes, with finitely
>many exceptions (Vinogradov, 1937); and 
>
>Every even number is the sum of a prime and a product of at
>most two primes, with finitely many exceptions (Chen, 1973,1978).

Maybe there was some confusion in the original poster's mind caused
by variant meanings of the phrase "almost all"?

It was realised early on by several authors that Vinogradov's result,
in the form that the number of ways an odd number can be expressed as
the some of three primes is asymptoticly equal to the "expected" number,
implies that almost all even numbers are sums of two primes, i.e.
if E(x) is the number of even numbers < x not so expressible, then
E(x) = o(x); indeed E(x) = O(x/(log x)^A) for all A. That was improved
to E(x) = O(x^c) for some c < 1 by Montgomery & Vaughan (Acta Arith.
27 (1975) 353-370).

But it's a long way from that that to "only a finite number of
exceptions", i.e. E(x) bounded.

Chris Thompson
Email: cet1 [at] cam.ac.uk
==============================================================================

From: Jan-Christoph Puchta <jcp@arcade.mathematik.uni-freiburg.de>
Subject: Re: Goldbach's Conjecture
Date: Wed, 26 Jul 2000 11:20:05 +0200
Newsgroups: sci.math
Summary: [missing]

Bob Silverman wrote:

> In article <snc0adqf6tt131@corp.supernews.com>,
>   "Gary Shannon" <reboot@rio.com> wrote:
>
> <snip>
>
> > BTW: I suppose that at best one could make it into a probabilistic
> argument
> > and prove that the conjecture is "probably true".  I wonder how close
> such a
> > probabilistic argument could be made to approach a probability of 1.0?
>
> Strictly speaking, this is nonsense. A theorem is either true or
> it isn't. One can't assign a probability of it being true.
>
> On the other hand,  the following result is known.
>
> Let S = {2n | 2n < X,  2n = p1 + p2  where p1, p2 are prime}
>
> It is known (via sieve methods) that the number of EXCEPTIONS is
> at most X^epsilon.  I believe the best proved value of epsilon so
> far is 1/4.  Someone please correct me if I am wrong.

I don't know if you are wrong, but if not I missed the greatest
mathematical event of the last 5 years. The first proof of epsilon<1 was
given by Montgomery and Vaughan (Acta Arith 27) in 1975, and although their
proof is constructive, it is really a nontrivial task actually to compute
their value. AFAIK, only in 1989 Jengrun Chen and Jianmin Liu proved
epsilon=0.95, and recently Li Hongze showed epsilon = 0.921 (Q. J. Math.
Oxford 50).

These results depend heavily on zerofree regions of Dirichlet L-serieses,
it is unlikely that they can be improved dramatically unless one finds some
q-analogue of Vinogradoff's method.

However, in special cases one can do better: Pintz showed that if n is
restricted to integers n, such that n=kp with p prime, k even and k<log n,
then there are at most x^{2/3+delta}exceptions to Goldbach's conjecture
among these numbers.


JCP
==============================================================================

From: Gerry Myerson <gerry@mpce.mq.edu.au>
Subject: Re: Goldbach conjecture
Date: Mon, 04 Sep 2000 11:17:59 +1000
Newsgroups: sci.math

In article <jgreene-A1BCDA.11310701092000@news.umn.edu>, John Greene 
<jgreene@d.umn.edu> wrote:

>     This is my understanding, as well, that the 10^20 assumes the ERH.  
> In a related matter, I have checked three primes up to a little past 
> 10^20 (assuming the two primes version up to 10^14) by finding a chain 
> of primes from 10^14 to 10^20, none differing from the others by more 
> than 10^14.  It only takes a little over a million primes, and took 
> about a week for Maple to crank out.  Yes, I did prove that each number 
> was, in fact, prime by supplying, for each p, an element of order p - 1.
> 
>     I'm told someone else did this calculation and published the results 
> (I have only a few megabites of primes and certificates, myself), but I 
> don't have a reference.

Some or all of these might be useful. 

Gerry Myerson (gerry@mpce.mq.edu.au)
*************************************
2000j:11143 11P32 (11Y35) 
Deshouillers, J.-M.(F-BORD2-ST); te Riele, H. J. J.(NL-MATH); Saouter, 
Y.(F-TOUL3-IR) 
New experimental results concerning the Goldbach conjecture. (English. 
English summary) 
Algorithmic number theory (Portland, OR, 1998), 204--215, 
Lecture Notes in Comput. Sci., 1423, 
Springer, Berlin, 1998. 

Summary: "The Goldbach conjecture states that every even integer $\geq 
4$ can be written as a sum of two prime numbers. It is known to be true 
up to $4\times 10\sp {11}$. In this paper, new experiments on a Cray 
C916 supercomputer and on an SGI computer server with 18 R10000 CPUs 
are described, which extend this bound to $10\sp {14}$. Two 
consequences are the following: (1) under the assumption of the 
generalized Riemann hypothesis, every odd number $\geq 7$ can be 
written as a sum of three prime numbers, and (2) under the assumption 
of the Riemann hypothesis, every even positive integer can be written 
as a sum of at most four prime numbers. In addition, we have verified 
the Goldbach conjecture for all of the even numbers in the intervals 
$[10\sp {5i},10\sp {5i}+10\sp 8]$, for $i=3,4,\cdots,20$ and $[10\sp 
{10i},10\sp {10i}+10\sp 9]$, for $i=20,21,\cdots,30$. 

"A heuristic model is given which predicts the average number of steps 
needed to verify the Goldbach conjecture on a given interval. Our 
experimental results are in good agreement with this prediction. This 
adds to the evidence of the truth of the Goldbach conjecture." 


\{For the entire collection see MR 2000g:11002.\}

                      Reviewed by Leszek Kaniecki 

98g:11115 11P32 (11Y11) 
Saouter, Yannick(F-RENNB-II) 
Checking the odd Goldbach conjecture up to $10\sp {20}$. (English. 
English summary) 
Math. Comp. 67 (1998), no. 222, 863--866. 

The author used his primality test and forty computers to prove that 
the maximal gap between consecutive primes smaller than $10\sp {20}$ is 
smaller than $4·10\sp {11}$. Using results in the paper of M. K. 
Sinisalo [Math. Comp. 61 (1993), no. 204, 931--934; MR 94a:11157] (all 
even numbers from the interval $[4,4·10\sp {11}]$ are sums of two 
primes) he obtains the following result: Every odd number greater than 
or equal to 7 is the sum of three prime numbers.


                  Reviewed by Leszek Kaniecki 


98f:11107 11P32 (11M26 11P55) 
Zinoviev, Dmitrii(1-OHS) 
On Vinogradov's constant in Goldbach's ternary problem. (English. 
English summary) 
J. Number Theory 65 (1997), no. 2, 334--358. 

The author shows by assuming the generalized Riemann hypothesis (GRH) 
that any odd number greater than $10\sp {20}$ can be written as a sum
of three primes. The proof is based on the circle method and the 
Riemann-Hadamard method which involves summation over the zeroes of
$L$-functions. Y. Saouter [Math. Comp. 67 (1998), no. 222, 863--866] 
proved recently that any odd number from the interval $[9,10\sp {20}]$ is
the sum of three odd primes. This implies that the odd Goldbach 
hypothesis is a consequence of GRH.

                  Reviewed by Leszek Kaniecki 

Cited in: 98g:11112 2000j:11144 1 715 106
==============================================================================

From: Mark Watkins <mwatkins@msri.org>
Subject: Re: Goldbach conjecture
Date: Sun, 03 Sep 2000 23:24:19 +0000
Newsgroups: sci.math

Gerry Myerson wrote:

>98f:11107 11P32 (11M26 11P55)
>Zinoviev, Dmitrii(1-OHS)
>On Vinogradov's constant in Goldbach's ternary problem. (English.
>English summary)
>J. Number Theory 65 (1997), no. 2, 334--358.
>
>The author shows by assuming the generalized Riemann hypothesis (GRH)
>that any odd number greater than $10\sp {20}$ can be written as a sum
>of three primes. The proof is based on the circle method and the
>Riemann-Hadamard method which involves summation over the zeroes of
>$L$-functions. Y. Saouter [Math. Comp. 67 (1998), no. 222, 863--866]
>proved recently that any odd number from the interval $[9,10\sp {20}]$ is
>the sum of three odd primes. This implies that the odd Goldbach
>hypothesis is a consequence of GRH.
>
>                  Reviewed by Leszek Kaniecki
>
>Cited in: 98g:11112 2000j:11144 1 715 106

Unfortunately, Zinoviev's proof contains some flaws.
He makes some numerical errors, such as his claim (by "elementary
computations" on page 348) that  phi(q)> 14.1 sqrt(q) for q>2330,
obviously incorrect for q=2520. On page 354, he says
"using phi(q)>=sqrt(q)", which fails for q=6. The third error
is in the first line of page 346, where the sum should be over
t=1,3,5,... instead of the even numbers; this affects his bound
in this part by about a factor of exp(pi/4).

Fortunately, Zinoviev's numerical bounds in other parts of the
paper are sloppy, and not very sharp. Upon bounding these parts more
sharply, and correcting the errors of above, one is actually able
to get the result claimed by Zinoviev.

===
Mark Watkins
mwatkins@msri.org
==============================================================================

From: Jan-Christoph Puchta <jcp@arcade.mathematik.uni-freiburg.de>
Subject: Re: Goldbach conjecture
Date: Tue, 19 Sep 2000 11:31:21 +0200
Newsgroups: sci.math

freelancefabulous@my-deja.com wrote:

> Actually it is pretty clear that the twin primes
> is a corollary of GC.
>
> Gary

Well, thinks seem to be more difficult. Although the standard approach
via lowe bound sieves applies equally well to prime twins and Goldbach,
e.g. Heath-Brown's conditional proof (If there are infinitely many
Siegel zeros, then there are infinitely many prime twins) does not apply
to Goldbach. Also by Turan gave an explicite formula for the number of
prime twins <= x, which involves only zeros of L-serieses with imaginary
part <6, however, Goldbach seem to depend on zeros with larger imaginary
part.


Jan-Christoph Puchta