Talk:Waring's problem

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Removed this:

Using the Hardy-Littlewood method?, g(k) can now readily be computed for all other values of k as well.

Really ? If we had to wait till 1986 to prove g(4)=19, I don't think we can readily compute g(k) yet. And if so, how ? Hardy & Littlewood died long before 1986. FvdP 20:06 Oct 23, 2002 (UTC)

I think this is simply wrong; no such method is known. Thanks for the catch. AxelBoldt 22:27 Oct 23, 2002 (UTC)

I guess we have to wait some more time, yes. Axel thank you too for catching the link to wrong Langrange's theorem. --XJamRastafire 00:48 Oct 24, 2002 (UTC)

This seems self-contradictory (emphasis is mine):

All other values of g are now also known, as a result of work by Dickson, Pillai, Rubugunday and Niven. Their formula contains two cases, and it is conjectured that the second case never occurs; in the first case, the formula reads g(k) = [...]

So is g known or only conjectured ? If it is known, what is Dickson (etc)'s second case about ? --FvdP 20:03 Feb 17, 2003 (UTC)

The exact value for g(k) is known for every k, in the sense that, if you give me k, I can give you g(k). Dickson's formula has two cases. In order to figure out which case of the formula to use for a particular k, I have to check a certain thing about the number k. If the outcome of that check is "YES", I pick the first case of the formula, if the outcome of the test is "NO", I pick the second case of the formula. Nobody has ever seen a value for k where the outcome was "NO". It is conjectured that there are no such values. Therefore I didn't list the second case of the formula in the article.

Maybe the sentence in the article should be reformulated somehow. AxelBoldt 21:46 Feb 17, 2003 (UTC)

OK, I understand it now. Thanks. My mathematical mind looks a bit rusty. Sad thing. --FvdP

Waring's problem really solved? I think not!

Shouldn't this page be fixed? I really really doubt that g(k) is known for all k now. Dickson's formula only gives conjectured values for g(k). His formula indeed has two cases, one of which is conjectured not to occur, but in any case the outcome of Dickson's formula can be easily computer for arbitrary k. This outcome however is *not* known to be the exact value of g(k). See for more information about the condition to be checked in Dickson's formula.

To repeat once more, the statement on this talk page that "The exact value for g(k) is known for every k, in the sense that, if you give me k, I can give you g(k)" is NOT true. What is true is that "if you give me k, I can give you the outcome of Dickson's formula". Dickson's formula gives only a conjectural value for g(k). Dickson's formula is from 1936. The value of g(4) was only proved to be 19 in 1986!

As this page is now, it gives the impression that the real values of g(k) are almost all known by Dickson's formula, and that only an unimportant secondary condition is left to be checked. This is the impression also when you search for Waring's problem on the web, since most information on the problem is now coming from this wikipedia article. So a fix is in order. I could probably do it, but I'd like to hear first what the original authors think of this.

The anonymous poster in this section obviously missed the requirement for k to be greater than or equal to 6 in Dickson, et. al.'s formula. In my opinion AxelBoldt's explanation above is correct. Does that mean that the anonymous comment above is "patent nonsense" and should be removed? TheGoblin 16:49, 26 January 2006 (UTC)
Hardy and Wright states that indeed g(k) is known for , but not at all that this was proved by Dickson. They describe it as an ongoing work, with the validity of Dickson's formula proved much later. In other words, Dickson's formula indeed only gave a conjectural value; but the conjecture now is proved. I think this means that the anonymous protest was a bona fide and not totally unreasonable misunderstanding.-JoergenB (talk) 16:49, 12 January 2008 (UTC)

- - - Vinogradov's bound was improved upon several times according to Ian Stewart (Game, Set and Math). J.R. Chen showed g(n)<= n(3logn+5.2). A more complicated improvement later by Mozzochi and Balasubramanian. Of more interest are the best known individual bounds for G(n) <=g(n), which from Stewart are for n=4,5,6...15: 3,4,7,8,34,32,102,51,135,150,166 and 181.Billymac00 13:49, 31 January 2006 (UTC)

Since G(4) is exactly 16 I think Billymac00's list of upper bounds for G(n) seems questionable. He is correct in that Vinogradov's bound has been improved several times... TheGoblin 22:09, 1 March 2006 (UTC)

formula for g(k)

the formula for g(k) is actually based on a very basic inequality. It is definitely a lower bound always, and the conjecture states that it also is enough. I changed it just before we discuss the formula.Evilbu 15:14, 4 February 2006 (UTC)

Everything is explained clearly here:

Who writes this stuff?

"Lagrange's four-square theorem was conjectured by Fermat in 1640"

as a special case of his Polygonal Number Theorem? That was 1638. When he proved it I don't know, perhaps two years later is correct, since it wasn't published. Relevant? Barely.

"and was first stated in 1621."

Really? Not by Diophantus?

"Apart from a certain ambiguity, all the other values of g are now also known, as a result of work by Dickson, Pillai, Rubugunday and Ivan M. Niven."

I have no idea if this is true or not. Mathworld states "Dickson (1936), Pillai (1936), and Niven also conjectured an explicit formula for g(s) for s>6". 18:58, 2 September 2006 (UTC)

Big oh

It is somewhat curious to write a formula with big Oh, and claim that it holds for every k. Kope 13:23, 11 July 2007 (UTC)


Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers contradicts g(4) = 19 [was established] in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouiller because Hardy and Littlewood both died before 1986 and they already knew from trivial computation that 79 requires 19 fourth powers, thus they knew that g(4) = 19. Icek (talk) 17:20, 23 November 2007 (UTC)

There is no contradiction. Let's reformulate the two claims. Call S the set of those natural numbers which cannot be written as the sum of 19 fourth powers. Hardy and Littlewood proved that S is finite and B-D-D proved that it is the empty set. This is much stronger and surely not identical. Kope (talk) 13:45, 26 November 2007 (UTC)
Sorry - somehow I unintentionally ignored the "sufficiently large" as it seemed unintuitive that smaller numbers would require more fourth powers - I have to be a bit more careful. Icek (talk) 18:39, 26 November 2007 (UTC)

Why analytic?

If Waring's problem is related to analytic number theory, can this link be made clear somehow? What analytic techniques are used in its solution? Thehotelambush (talk) 02:00, 12 November 2008 (UTC) Large quantities of calculus and complex numbers are needed to answer your question. —Preceding unsigned comment added by (talk) 15:11, 8 August 2009 (UTC)


Hmm, I am no math wiz or anything, but, the beginning paragraph seems off... does waring's problem really state "19" fourth power numbers? that seems totally random, if it does >.> why wouldn't it be 16? (talk) 00:46, 13 January 2010 (UTC)

Read down to the first paragraph of the next section. How do you express 79 as a sum of only 16 fourth powers? —David Eppstein (talk) 00:50, 13 January 2010 (UTC) seems to have little or no formal education. —Preceding unsigned comment added by (talk) 16:24, 28 September 2010 (UTC)

Changes on 14th October


I made the changes on 14th October. There were quite a number of obscurities and oversights in the article which I have attempted to correct. For example there was a misleading reference to Euler which should be to his son. The article omitted to mention Vinogradov's important paper of 1959, but mentions Karatsuba's as if it were the breakthrough. p-adic ideas were first introduced in the late 1930s by Davenport as a refinement of the Hardy and Littlewood diminshing ranges technique. In somewhat different ways they were used by Vinogradov and Linnik. J. A. Euler's formula is proved for k <= 471,000,000, and so I added chapter and verse on this. There were quite a number of other things like this. I added a reference to the survey article of Wooley and myself. This is an area where there has been a massive amount of work in since 1920 and the article can only really act as a guide to further reading. If anyone would like to discuss the changes through this talk page please alert me at

Bob Vaughan

Rvaughan2000 (talk) 18:57, 16 October 2010 (UTC)


Hi, I don't understand the following sentence: "Davenport showed that G(4) = 16 in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12)."

Which of the two 14's mentioned is reduced to 12? If it is the first this doesn't seem to be an improvement. If it is the second this would imply that G(4) = 14, contradicting the statement later in the article that each number of the form 31*16^n requires 16 fourth powers. Ok, now that I am typing this I can think of a third possibility where both 14's are simultatiously lowered to 12. That is probably what is meant but if so the sentence should perhaps be rewritten to make this clearer.

Apart from that: showing that every sufficient large number congruent to 1 through 14 mod 16 can be written as a sum of 14 fourth powers only seems to show that G(4) is at most 16, so maybe the remark about 16 being a lower bound as well (which we find further down the article) should be moved up to this point in the article.

Vincent — Preceding unsigned comment added by Octonion (talkcontribs) 20:24, 5 February 2012 (UTC)

Wooley then established that for some constant C,[20]

[20] The Hardy-Littlewood method, R. C. Vaughan, 2nd ed., Cambridge Tracts in Mathematics, CUP, 1997

I don't see Wooley. Where did he prove his great result? ru:Участник:МетаСкептик12

There is a detailed answer in the Zentralblatt review: I have added a link to the citation in the article. Deltahedron (talk) 16:29, 25 July 2012 (UTC)


The article currently[1] states, "Prior to the posing of Waring's problem, Joseph-Louis Lagrange theorized that any positive integer could be represented as the sum of four perfect squares". "Theorized" is not a word that is used in mathematics: what does it mean? It seems that it must mean either "conjectured" or "proved" but, from the chronology, it's impossible to tell which: Waring posed his problem in 1770 and Lagrange proved the four-square theorem in the same year. I would guess that Waring posed the problem after hearing about Lagrange's proof. On the other hand, it's also possible that Waring posed the general problem and Lagrange heard of this and proved the square case. Does anyone know what actually happened? Dricherby (talk) 10:51, 29 October 2013 (UTC)