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 wrote: >"Mark Lamb" 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 Subject: Re: Goldbach's Conjecture Date: Wed, 26 Jul 2000 11:20:05 +0200 Newsgroups: sci.math Summary: [missing] Bob Silverman wrote: > In article , > "Gary Shannon" wrote: > > > > > 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 Subject: Re: Goldbach conjecture Date: Mon, 04 Sep 2000 11:17:59 +1000 Newsgroups: sci.math In article , John Greene 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 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 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