Goldbach Conjecture and Schnirelmann's "300,000 primes"
mwat...@maths.ex.ac.uk
I'm seeking clarification on an oft-sited claim that Lev Schnirelmann
proved in 1931 (or 1939?) that every even integer can be expressed as
a sum of of not more than 300,000 primes.
The 300,000 always stuck me as a bit strange - it's hard to imagine
how number theoretical reasoning would arrive at a number like that.
Mathworld gives as its reference
Dunham, W. Journey through Genius: The Great Theorems of Mathematics.
New York: Wiley, p. 83, 1990.
which references this back to H. Eves book on the history of
mathematics.
Mathworld also cites
Schnirelman, L. G. Uspekhi Math. Nauk 6, 3-8, 1939.
which was presumably posthumous, as Schnirelmann reportedly commited
suicide in '38.
H. Eves gives the year as 1931 (but he also wrongly states the year of
Schnirelmann's death as '35).
To confuse matters further, Wikipedia, without citation, claims
"In 1930, Lev Schnirelmann proved that every even number n ‰?´ 4 can be
written as the sum of at most 20 primes."
The 20 here seems a lot more plausible... So where did the 300,000
come from? It almost seems like a mathematical in-joke that's
accidentally got into circulation.
Can anyone who's actually seen the original 1930/31/39(?) paper by
Schnirelmann and can clear this up for me?
Thanks!
MW
Olivier
Hi,
Why don't you use decent material like math reviews or zentralblatt?
And read papers on the matter, like Klimov, Shapiro & Warga,
Deshouillers, Vaughan, Riesel & Vaughan, Ramar? ?
I've never seen in any decent litterature the 300 000 you quote.
It is definitely not hard to see how mathematical reasoning
would arrive at a number like that, just remember that Brun
sieve uses estimates of sum 1/p and that a small offset on them
is easy but has tremendous consequences. Selberg sieve simplified
matters, but only from 1940/50 onwards.
Ricci got 67 in 1937 for the *asymptotic* S'nirel'man constant, which
Shapiro & Warga improved to 20,
which Riesel & Vaughan reduced in 19 which is the limit of the
method since one looses a factor 8 at least in the bound
on the number of representations as a sum of two primes.
This I improved to 6 (5 for odd numbers). Since this is unpublished,
here is the proof:
(1) show that the density of sums of two primes
is > 1/ (2 x (2+epsilon)) by using my method.
(2) show that the density of sums of three primes is
> 1/(2 (2-epsilon)) by an essential components argument.
(3) add both sequences.
It is even possible to reach 3 for odd numbers, asymptotically,
and some people are indeed trying to use that to show that
every odd integer is a sum of three primes, but I leave them
the pleasure to do so. Anyway: we now know how to push
S'nirel'man's method by adding Shapiro and Warga kind of information
to show that every odd large enough number is a sum of a most
three primes. Though the link with S'nirel'man's method
is now quite hard to see and not everyone would say this
still belongs to this line of approach:p.
The earlier bound (for the full S'nirel'man constant) I know
is 2.10^{10} by Êeptickaja in 1963
(
unpublished, see
MR0205959 (34 #5784)
€?udakov, N. G.; Klimov, N. I.
Concerning the Ênirelæ1man constant. (Russian)
Uspehi Mat. Nauk 22 1967 no. 1 (133), 212--213
I quote the full reference of this paper
since here is what Warga writes about it
"The authors list several books on the theory of numbers in which it is
alleged that every integer $n\geq N_0=2$ can be represented as a sum of
$S=20$, or less, primes. These books misquote results of H. N. Shapiro
and the reviewer [Comm. Pure Appl. Math. 3 (1950), 153--176; MR0037323
(12,244c)], who proved by elementary means that every sufficiently large
integer $n$ can be so represented. The authors state that the best
explicit estimates are, at the present time, those of A. A. Êanin and T.
A. Êeptickaja (presented at the Fourth Sci. Conf. Math. Depts. Ped.
Inst. Volga Region 1963, apparently unpublished), who proved that
$S=2\cdot 10^{10}$ for $N_0=2$, and those of K. G. Borozdkin [Proc.
Third All-Union Math. Conference (Moscow, 1956), Vol. I, p. 3, Izdat.
Akad. Nauk SSSR, Moscow, 1956], who proved that $S=4$ for
$N_0=\exp(\exp(16.038))$."
)
and the first published is 6. 10^9 by N.I. Klimov.
I have not read the russian paper,
but Deshouillers reads russian, studied this problem
and has told me it was according to him the earliest
estimate known.
Cheers,
Amities,
Olivier
mar...@gmail.com
Vinogradov's result about three primes is not satisfactory for some
reason?
> This I improved to 6 (5 for odd numbers). Since this is unpublished,
> here is the proof:
> (1) show that the density of sums of two primes
> is > 1/ (2 x (2+epsilon)) by using my method.
> (2) show that the density of sums of three primes is
> �> 1/(2 (2-epsilon)) by an essential components argument.
> (3) add both sequences.
This is not a proof. Only Step 3 seems clear (it follows from
Shnirelman's statement).
Mark
mwat...@maths.ex.ac.uk
Thanks for the clarification!
Incidentally, the Schnirelmann paper which I mentioned as the source
for the "300000" claim
[Uspekhi Math. Nauk 6, 3-8, 1939]
turns out to be his obituary.
I found a PDF of the Russian original here:
http://tinyurl.com/yzzk9ln
If someone who can read Russian can be bothered, I'd be interested to
know if it does actually make this claim. I'm interested in this more
from an historical, rather than a number theoretical, perspective.
Thanks,
MW
dro...@gmail.com
I don't think the obituary makes the claim about 300,000. i.e., the
claim that every even number can be
represented as a sum of at most 300,000 primes. In a couple of days
I'll make the translation of the obituary available
on line, (it is 6 pages long), but as far as I can tell there is no
mention about the number 300,000 there. What it says about the
Goldbach conjecture is that there a fixed number X, such that every
sufficiently large integer (even or odd)
can be expressed as a sum of at most X primes, but there is no mention
of the value of X. The way Schnirelmann
shows that is to prove that the set {p+q | p, q primes} has a positive
density.
As ever,
Vlad
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
* Vladimir Drobot
* Retired and gainfully unemployed
* http://www.vdrobot.com
* mailto:dro...@pacbell.net
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
Olivier
>> It is even possible to reach 3 for odd numbers, asymptotically,
>
> Vinogradov's result about three primes is not satisfactory for some
> reason?
This is a methodological problem.
Say we want to prove it by using only the prime number theorem
in ap for a finite number of ap's.
>> (1) show that the density of sums of two primes
>> is > 1/ (2 x (2+epsilon)) by using my method.
I can't be more precise without writing it out.
I proved that this density is > 1/ (2 x (2.48))
and somehow explained there why the 2.48 was a 2+epsilon
for any positive epsilon depending on the strength
of your computer.
See section 2 of
http://math.univ-lille1.fr/~ramare/Maths/Article.pdf
>> (2) show that the density of sums of three primes is
>> > 1/(2 (2-epsilon)) by an essential components argument.
(See also Khintchin 1933) Erdos Theorem, improved for
primes by Ruzsa, see also Plunnecke (and the Ruzsa
proof of this result). Look at Halberstam & Roth book
"sequences" and at Ruzsa's papers on essential components.
An additive basis is an essential component.
P. Erdos
On the arithmetical density of the sum of two sequences one of which
forms a basis for the integers. (English)
Acta arith., Warszawa, 1, 197-200 (1936)
http://matwbn.icm.edu.pl/ksiazki/aa/aa1/aa123.pdf
Rohrbach, H.
Einige neuere Untersuchungen �ber die Dichte in der additiven
Zahlentheorie. (German)
Jber. Deutsche Math.-Verein. 48, 199-236 (1939).,
http:/resolver.sub.uni-goettingen.de/purl?GDZPPN002132443
Ruzsa, I. Z.
On an additive property of squares and primes. (English)
Acta Arith. 49, No.3, 281-289 (1987).
http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4936.pdf
Ruzsa, I. Z. (1987). "Essential components". Proceedings of the London
Mathematical Society 54: 38�56.
Ruzsa, Imre Z.
An application of graph theory to additive number theory. (English)
Sci., Ser. A 3, 97-109 (1989)
>> (3) add both sequences.
>
> This is not a proof. Only Step 3 seems clear (it follows from
> Shnirelman's statement).
Depends to whom :p
As a matter of fact, in this precise case, this is not a Theorem
of Schnirelman but of sorry-I-dont-remember-who: when the
asymptotic density of A is alpha and the one of B is beta
and alpha+beta>1 then A+B contains every integer from some
point onwards. This is easy to prove. And if I remember
Mann's proof correctly, it starts with a
"we can assume alpha+beta <= 1 for otherwise ..."
Again in Halberstam & Roth book. This has to do with
asymptotic densities but it can readily be made explicit.
By the way: Schnirelman or S'nirel'man (though I tend to
believe now S'nirelman should be more accurate), but no Shnirelman.
Ok, I've worked enough:)
HTH, A.O.
tc...@lsa.umich.edu
mwat...@maths.ex.ac.uk <mwat...@maths.ex.ac.uk> wrote:
>I'm seeking clarification on an oft-sited claim that Lev Schnirelmann
>proved in 1931 (or 1939?) that every even integer can be expressed as
>a sum of of not more than 300,000 primes.
"The Book of Prime Number Records" by Ribenboim gives the number 800,000
(instead of 300,000) and implies that Schnirelmann calculated it, but
doesn't give an explicit reference for the 800,000 figure. In "The New
Book of Prime Number Records," the 800,000 figure is still there, but
the sentence has been reworded and no longer suggests that Schnirelmann
stated this number himself.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
mwat...@maths.ex.ac.uk
> In article <hipp0r$2uh...@dizzy.math.ohio-state.edu>,
>
> mwatk...@maths.ex.ac.uk <mwatk...@maths.ex.ac.uk> wrote:
> >I'm seeking clarification on an oft-sited claim that Lev Schnirelmann
> >proved in 1931 (or 1939?) that every even integer can be expressed as
> >a sum of of not more than 300,000 primes.
>
> "The Book of Prime Number Records" by Ribenboim gives the number 800,000
> (instead of 300,000) and implies that Schnirelmann calculated it, but
> doesn't give an explicit reference for the 800,000 figure. �In "The New
> Book of Prime Number Records," the 800,000 figure is still there, but
> the sentence has been reworded and no longer suggests that Schnirelmann
> stated this number himself.
I've just received a translation of the Schnirelmann obituary from
Uspekhi Math. Nauk 6, 3-8, 1939 and it certainly makes no mention of
300,000 (or any other number) in this connection. It's a complete
mystery where Howard Eves got this from. Perhaps someone should ask
Ribenboim where he got the 800,000 claim from.
MW
dro...@gmail.com
<mwatk...@maths.ex.ac.uk> wrote:
>> Incidentally, the Schnirelmann paper which I mentioned as the source
>> for the "300000" claim
>> [Uspekhi Math. Nauk 6, 3-8, 1939]
>> turns out to be his obituary.
>> I found a PDF of the Russian original here:http://tinyurl.com/yzzk9ln
>> <. . . .>
>> If someone who can read Russian can be bothered, I'd be interested to
<dro...@pacbell.net> replied
> I don't think the obituary makes the claim about 300,000. i.e., the
> claim that every even number can be
> represented as a sum of at most 300,000 primes. In a couple of days
> I'll make the translation of the obituary available
As promised, I provide a translation of the above reference. You can
see it or download it from
http://www.vdrobot.com/Handouts/SchnrlmnObit.pdf
Some notes. The Russian version of this obituary first appears in Rec.
Math. [Mat. Sbornik] N.S., 4(46):1 (1938), I-III. The reference above
is
essentially a reprint, with minor changes. Poke on Google for
"Schnirelmann obituary" to see it there.
I could not find who was the author, but I really didn't spend a lot
of time on it.
There is a very extensive article (in Russian) about Schnirelmann, by
V. Tikhomirov and V. Uspenski.
It appears in the publication Kvant in 1996, for the details see
http://kvant.mirror1.mccme.ru/1996/02/lev_genrihovich_shnirelman.htm
It is about 20 - 25 pages long, going into various aspects of his life
and his mathematics in a much more detail than the above, which is a
rather formal and dry obituary.
It is an extremely interesting an lively article, and I think it would
be of great interest to the English speaking mathematical community.
One of the topics, for example, is the discussion of his death. He
apparently committed a suicide, at the age of 33, and the article
discusses
possible motives - an interest of NKVD (precursor to KGB) in
Schnirelmann being one of them. Being retired and gainfully
unemployed, with lots of time,
I'd be willing to provide a translation, but before I start I'd like
to know if one already exists, perhaps. I am not into re-inventing
wheels.
I poked around the web, and could not find anything, but if anyone
knows that a translation was already done, let me know. Otherwise,
I'll start plodding.
Heine Rasmussen
I have done some searches myself, and I think it is extremely unlikely
that there is an English translation of this article on the web (or
anywhere else). So go ahead! He must have been quite a character, and
I look forward to read the translation.
JEMebius
> [...]
>
> I have done some searches myself, and I think it is extremely unlikely
> that there is an English translation of this article on the web (or
> anywhere else). So go ahead! He must have been quite a character, and
> I look forward to read the translation.
>
Perhaps mr Matthew Watkins can help: see his message at The Math Forum @ Drexel at URL
http://mathforum.org/kb/message.jspa?messageID=6956093&tstart=15
BTW, I did a Google search for 'obituary "lev shnirelmann"' with the name intentionally
misspelled. With misspellings of search terms one often finds additional information.
Good luck: Johan E. Mebius
dro...@gmail.com
As promised, I offer here a translation of an article about L. G.
Schnirel'mann, written by
V. Tikhomirov and V. Uspenskii. It appeared in the Russian magazine
Kvant in 1996, No 2. pp. 2-6. The
translation took longer than I expected, (for various reasons, mainly
laziness), but here it is.
The article is ostensibly written for high school students, and in a
good year there maybe 5 such students on
the planet who would understand and follow most of it. But, it can be
read with interested and profit by a
typical Ph. D. and there is even a suggestion to a wannabe Ph. D. for
a thesis topic. (Removal of the
"twice differentiable" hypothesis from one of the Schnirel'mann's
theorem) There is a direct
connection with this thread in this group as follows: The
Schnirel'mann introduced a method
which allows one to show that there is a number C with the property
that any even number can be written
as a sum of at most C primes. Goldbach conjecture states that C = 2. A
question was raised as to the value of
C that results from the Schnirel'mann's method. A number 300,000
appears in several places, but
no one could come up with an exact reference to an honest calculation.
The present article perpetuates
the myth, in a way, it states that C is of the order of several
hundreds of thousands, but no
reference is given again.
The article also provides an interesting glimpse into the private
lives of Soviet mathematicians:
Luzin is alleged to have been a mystic, and Schnirel'mann himself
committed suicide at the age of 33 because
he expected to be arrested and questioned (tortured?) by NKVD (the
precursor of the KGB).
The URL address is:
http://www.vdrobot.com/Handouts/SchnrlmnnUsenet.pdf