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 更正：哥德巴赫猜想和陈氏定理 |
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作者：Anonymous 在 罕见奇谈 发贴, 来自 http://www.hjclub.org
哥德巴赫猜想有许多等价命题。楼下拙文（陈景润的故事简述）所据的是一本中文通俗读物。为防止误导网友，现再贴上从网上找出的资料，其中“1+1”的提法与本人文中的不尽相同。关与陈景润的贡献
“1+2”，下文也有提及. 看来，本人文中的“合数”应代之以“大于2的偶数”。又陈氏所得结果至今无人超过。
Goldbach wrote a letter to Euler dated June 7, 1742 suggesting (roughly) that every
even integer is the sum of two integers p and q where each of p and q are either
one or odd primes.?Now we often word this as follows:
Goldbach's conjecture: Every even integer n greater than two is the sum of two
primes.
This is easily seen to be equivalent to
Every integer n greater than five is the sum of three primes.
There is little doubt that this result is true, as Euler replied to Goldbach:
That every even number is a sum of two primes, I consider an entirely certain
theorem in spite of that I am not able to demonstrate it.
Progress has been made on this problem, but slowly--it may be quite awhile before
the work is complete.?For example, it has been proven that every even integer is
the sum of at most six primes (Goldbach suggests two) and in 1966 Chen proved
every sufficiently large even integer is the sum of a prime plus a number with no
more than two prime factors (a P2).
Vinogradov in 1937 showed that every sufficiently large odd integer can be written
as the sum of at most three primes, and so every sufficiently large integer is the
sum of at most four primes.?One result of Vinogradov's work is that we know
Goldbach's theorem holds for almost all even integers.
Various heuristic estimates are avaliable for the number of solutions there should
be to 2n = p + q (with p, q prime). And these of course grew in size with n.
Recently, Jean-Marc Deshouillers, Yannick Saouter, and Herman te Riele have
verified this up to 1014 with the help of a Cray C90 and various workstations.?If we
will accept the Riemann Hypothesis, then this is enough to prove the odd Goldbach
conjecture: Every odd integer greater than five is the sum of three primes.
Descartes also was aware of the two prime version of Goldbach's conjecture before
Goldbach was.?So is it misnamed??Erd歴 said that it "is better that the conjecture
be named after Goldbach because, mathematically speaking, Descartes was infinitely
rich and Goldbach was very poor."
When verifying the Goldbach conjecture for n we quickly see that it is very easy to
find many primes which add to n.?In 1923 Hardy and Littlewood took the first
major step toward the proof of the Goldbach conjectures using their circle method
[HL23].?Among other things, they conjectured that the number of ways of writing n
as the sum of two primes, G(n), is asymptotic to twice the twin prime constant times
n/(log n)2 times the product of (p-1)/(p-2) taken over the prime divisors p of n.
That is a lot of solutions for a large n so there will be a solution with one of the
prime quite small.?For example, when checking all n up to 100,000,000,000,000,
Richstein [Richstein2000] found the smaller prime is never larger than 5569 used in
the following partition:
389965026819938 = 5569 + 389965026814369.
It was conjectured [GRL89] that the smallest prime in a partition of n might be
bounded by a constant times (log n)2 log log n.?Richstein's work shows that the
constant 1.603 (from n=6) sufficies for n < 4.1014.
See Also: odd Goldbach conjecture
Related pages (outside of this work)
* Prime conjectures
* Verification of Goldbach's conjecture to 4.1014 (and a list of champions)
References:
DRS98
J. M. Deshouillers, H. J. J. te Riele and Y. Saouter, New experimental results
concerning the Goldbach conjecture.?In "Proc. 3rd Int. Symp. on Algorithmic Number
Theory," LNCS volume 1423, pp. 204--215, 1998.
FR89
Fliegel, H. F. and Robertson, D. S., "Goldbach's comet: the numbers related to
Goldbach's conjecture," J. Recreational Math., 21:1 (1989) 1--7.
GRL89
A. Granville, H. J. J. te Riele and J. van de Lune,"Checking the Goldbach conjecture on
a vector computer" in Number theory and its applications.?R. A. Mollin editor,
Kluwer, Dordrect, pp. 423--433, 1989.
HL23
G. H. Hardy and J. E. Littlewood, "Some problems of `partitio numerorum' : III: on
the expression of a number as a sum of primes," Acta Math., 44 (1923)
1-70.?Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561-630, Clarendon
Press, Oxford, 1966.
Ramare95
O. Ramar? "On Schnirelmann's constant," Ann. Sc. Norm. Super. Pisa, 22:4 (1995)
645-706.
Ribenboim95
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag,
New York, NY, 1995. (Annotation available)
Richstein2000
J. Richstein, "Verifying the goldbach conjecture up to 4?1014," Math. Comp., 70
(2001) 1745--1749. (Abstract available)
Sinisalo93
M. Sinisalo, "Checking the Goldbach conjecture up to 4?1011," Math. Comp., 61:204
(1993) 931--934.
Wang84
Y. Wang editor, Goldbach conjecture, Series in Pure Mathematics volume 4, World
Scientific, Singapore, 1984. (Annotation available)
又据博讯消息，称广东一农民近日彻底证明了哥德巴赫猜想“1+1=2”，恐怕从未有数学家听到过
这个命题。
作者：Anonymous 在 罕见奇谈 发贴, 来自 http://www.hjclub.org |
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