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 http://mathworld.wolfram.com/WaringsProblem.html 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 $k\geq 6$, 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: http://www.mathpages.com/home/kmath316.htm

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". 59.112.47.213 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)

g(4)

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 81.148.89.37 (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?173.28.10.216 (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)

173.28.10.216 seems to have little or no formal education. —Preceding unsigned comment added by 93.97.194.200 (talk) 16:24, 28 September 2010 (UTC)

I sympathize with the first comment, however little formal education the commenter may or may not have had. Waring's conjecture, as given by various sources, was that “Every number is the sum of 4 squares; every number is the sum of 9 cubes; every number is the sum of 19 fourth powers; and so on." The first two numbers appear to show a pattern: 4 is the square of 2, and 9 is the square of 3, so if the apparent pattern continues we might expect the next number to be 16, the square of 4, but in fact it is given as 19. It would be reasonable to suspect a typo or printer's error (setting a 6 upside down to give 19 instead of 16). I accept that 19 is indeed correct, but it breaks the apparent pattern and seems to come out of the blue. Perhaps there is no pattern after all. But if so, what did Waring mean by 'and so on'? This seems to imply not only a continuing pattern, but a continuing pattern which ought to be obvious to the reader. Not to me, anyway. Added: see my comment 'What Waring said' below. This rules out the possibility of a misprint. 2A00:23C8:7907:4B01:94C5:5A34:F356:3D09 (talk) 17:32, 15 June 2022 (UTC)

It sometimes helps to actually read a text before commenting on it. The conjectural value 2^k + [1.5^k] - 2 for g(k) is explained in the second paragraph of the article, and does provide a nice pattern. --Sapphorain (talk) 15:32, 19 June 2022 (UTC)

Changes on 14th October

Hi,

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

rvaughan@math.psu.edu

Bob Vaughan

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

G(4)

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 (talk • contribs) 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)

"Theorized"

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)

John Conway

This article should mention the fact that John Conway found g(5) while he was an undergraduate. 118.210.185.246 (talk) 11:37, 13 March 2014 (UTC)

Copy Editing

• I removed "there" because it is not grammatically necessary.
• I changed "... such that ..." with "such ... that ..." because the former is intuitive to people knowing the mathematical "such that" symbol (cursive E) whereas the latter is more intuitive to laypeople, for whom Wikipedia is written.
• I removed the parenthetical example because its length necessitates a separate clause.

Duxwing (talk) 16:38, 5 April 2014 (UTC)

You changed "whether ... there exists X" to "whether ... exists X". This is ungrammatical. "Whether X exists" might be acceptable, but in this case X is such a long phrase that readers would be left for a long time wondering what the verb is. As for whether the second part of the sentence is separated by parens or a semicolon, it seems very unimportant to me; better would have been to make it a separate sentence. —David Eppstein (talk) 16:46, 5 April 2014 (UTC)

• Why is "there exists" grammatical whereas "exists" alone is not? Perhaps "for every X there is a Y" generally is tortured and should be rewritten "a Y exists for every X".
• You have not mentioned whether "... such that ..." is better written "such ... that ...": are we therefore agreed that it is?
• The example is subordinate to the definition and therefore should follow a semicolon.

Duxwing (talk) 02:21, 6 April 2014 (UTC)

I'm not exactly sure how to diagram the grammar of "there exists" but it is extremely standard usage in mathematics. "Exists X" is ungrammatical because in English we put the subjects before the verbs.Your use of "such" instead of "such that" seems also strange and unidiomatic to me. Again, "such that" is extremely standard in mathematics. —David Eppstein (talk) 02:31, 6 April 2014 (UTC)

*"For Y exists X" is grammatical because English verbs can precede their subjects; e.g., the famous slogan, "Drill, baby, drill!" or the advertising phrase, "For all your X needs, we have Y".

*Wikipedian articles and especially their leads are written for the lay, for whom "such ... that ..." is more intuitive than "... such that ..." because the former is standard English whereas the latter uses a term-of-art.

Duxwing (talk) 03:24, 6 April 2014 (UTC)

"Drill, baby, drill" is imperative. The rules are different for that mode, and the subject is the implied "You", which is not stated at all so it doesn't come after the verb. "We have Y" has its subject, "We", before its verb. And using standard mathematical phrasing is a better idea than making up your own nonstandard phrasing. In this particular case, it's also important to understand that the phrasing "for every X there exists Y such that every Z has property P" is the prose version of a precise mathematical statement using logical quantification symbols, of the form $\forall X\exists Y\forall ZP(Z)$, and that small changes to the text can introduce mathematical errors in that interpretation. —David Eppstein (talk) 04:08, 6 April 2014 (UTC)

Pondering this problem last night, I noticed your objection that verbs must follow their subjects also applies to the original lead because "there" is not the sentence's subject: it is a preposition. If it is a preposition, then the preceding prepositional phrase "for..." renders it redundant by already defining the argument's domain. This redundancy is general, also applying to sentences like "Alexander the Great, he was a brilliant tactician," which should be written "Alexander the Great was a brilliant tactician". Therefore, you cannot demand that "there" remain by claiming its grammatical necessity.

This conclusion also holds in formal notation because "there" also is a pronoun for the domain's antecedent definition, which is included in the sentence and therefore requires no pronoun. The original sentence roughly formally noted therefore would be

Where X is true

Let Y refer to "X is true"

Where Y is true, Z

This notation could be transitively reduced to "Where X is true, Z". Also, even if "there" were mathematically necessary, its presence and the following wordy phrase would only confuse our lay readers.

Moreover, my notation is not made up; it exists throughout English in sentences like 'in this valley flows a river'. These sentences can become independent clauses in larger sentences like "In this valley flows a river, whose deep blue waters sparkle under the noon sun" or logical statements like "If in this valley flows a river, then cross it with a bridge". I would only thus write sentences wherein the noun or noun phrase following the verb is long; e.g., "for every natural number K exists such a positive integer S that every natural number is the sum of...".

Finally and perhaps fundamentally, the definition is wordy. I cannot parse its language, which, most charitably interpreted, seems to be written by mathematicians for mathematicians. Duxwing (talk) 18:59, 6 April 2014 (UTC)

The phrase complained of appears to conform to standard mathematical English conventions, namely, "for all/each/every X there exists (a/an/) Y such that P(X,Y)". It would be quite unconventional to omit "there". The example "for every natural number K exists such a positive integer S that every natural number is the sum of..." does not sound like either mathematical or idiomatic English. Deltahedron (talk) 19:12, 6 April 2014 (UTC)

Oh, hello! :) I have long known the "For X exists Y" to be standard, whereas I have not known "there exists" to be standard, often encountering "there is" or "exists" or "is"; would you please show me where I can learn about standard phrasing? Omitting "there" would be conventional English because "there" is redundant because it literally immediately follows the domain whereto it refers, and writing "such ... that ..." is conventional because "such" describes what follows and "that" describes the effect. Duxwing (talk) 01:35, 7 April 2014 (UTC)

Here are a couple of references describing this specific turn of phrase ("there exists").: [2] [3]. More generally for mathematical writing you might also find Knuth's book [4] helpful. —David Eppstein (talk) 03:33, 7 April 2014 (UTC)

Another few: [5], [6], [7]. Deltahedron (talk) 06:24, 7 April 2014 (UTC)

Wow! Thanks! :) I will consult these daunting texts when confused. If we want to be accurate without being excessively technical, then an idiomatic defintion in the lead and a symbolic one in the first section would respectively inform the lay and satisfy the learned. I suggest that this idiomatic definition be my afore-suggested one: "For ... exists such a ... that ...". Duxwing (talk) 01:56, 8 April 2014 (UTC)

As we have already said repeatedly, that word ordering is unidiomatic and nonstandard. I suggest we not do that. —David Eppstein (talk) 03:10, 8 April 2014 (UTC)

Its being non-standard should not matter in the lead because the lead is not written for laypeople, who would not understand its being non-standard and might be confused by "standard" (no scare-quotes intended) phrasing. Why is my phrasing unidiomatic? The sentence structure I suggest works in other sentences like "in the valley flows a river," wherein the verb precedes the noun, and like "He caused such a ruckus that the neighbors awoke," wherein "such ... that ..." is the structure. I therefore see no general reason to keep "there" or use the awkward "such that" in the lead, wherein readability must supersede precision lest lay readers should be driven away. To exemplify unto absurdity, consider non-mathematical sentences written to this technical standard: "in the valley there flows a river" could be a run-on and has an unclear prepositional phrase, and "he caused a ruckus such that the neighbors awoke" is tortured by a linguistic tick.

The lead's linguistic ticks transform it from a lay-readable paragraph to a technical journal entry requiring evidently voluminous reading for its first sentence's grammar alone. This transformation and its requirements thus violate sections 2.7.7 and 2.7.8 of Wikipedian policy "What Wikipedia is Not," which state that:

• 2.7.7 "A Wikipedia article should not be presented on the assumption that the reader is well versed in the topic's field. Introductory language in the lead (and also maybe the initial sections) of the article should be written in plain terms and concepts that can be understood by any literate reader of Wikipedia without any knowledge in the given field before advancing to more detailed explanations of the topic. While wikilinks should be provided for advanced terms and concepts in that field, articles should be written on the assumption that the reader will not or cannot follow these links, instead attempting to infer their meaning from the text."
• 2.7.8 "Academic language. Texts should be written for everyday readers, not for academics. Article titles should reflect common usage, not academic terminology, whenever possible."

Following this policy, I have suggested changing the lead's "academic language," which requires being "well versed in the topic's field" to understand, to the aforementioned "plain terms and concepts," which technical phrases like "there exists" or "... such that ..." evidently are not because I, a layperson, could not understand them. Knowing that accuracy is important, I have suggested also moving this technical description somewhere it will not confuse lay readers; e.g., a sub-heading titled "Formal Definition" including technical English and symbolic notation. Duxwing (talk) 04:25, 8 April 2014 (UTC)

• This is not the way things work at present. There are numerous ways of expressing the formula $\forall XexistsY$ in idiomatic English, such as, "for every X there is a Y" or "there is a Y for each X" or "for all X there exists Y". Mathematical convention is to use the last of these, which is acceptable English if somewhat stilted. To omit "there", as in "for all X exists Y" is not idiomatic English. Arguing by analogy with other constructions is beside the point. Of the two sentences

For every man in this world, there is a woman ready to marry him.

For every man in this world, exists a woman ready to marry him.

only the first is acceptable English.

As for the academic language point, I have explained that the "for all ... there exists" is a choice of idiomatic English. I frankly find it very hard to believe that a native speaker of English cannot understand the sentence "For all X there exists Y" but can readily understand "For all X exists Y".

A "literate reader" of an article must be presumed to understand the basic terminology of the subject and it is absurd to suggest that articles would or could be written any other way. It would be ridiculous to suggest that an article on biology could not use terms such as "genus" or "species" or "virus" or "animal" in the introduction, or in physics the word "matter" or "space" or "time" or "energy", or in chemistry the words "molecule" or "atom" or "element" or "compound". All of these are technical terms in the various subjects. Do you really expect an article on mice to begin "A mouse is a small furry thing that moves about and eats things that don't move about, belonging to a large group of quite similar moving things which contains some smaller groups of rather more similar small furry moving things."

Rewriting the Manual of Style to support your comments here is not the way to resolve this discussion. Deltahedron (talk) 21:25, 10 April 2014 (UTC)

arXiv is not a reliable source

Sorry, but arXiv is not a reliable publication: it contains only preprint-type papers, none of which is peer-reviewed. It is not an acceptable source for an encyclopedia. I am perfectly aware that Trevor Wooley is a more than decent mathematician, who has published a number of good papers in prestigious journals. But, precisely for this reason, if his results are correct they will eventually be published in a regular, peer-reviewed journal: it is only then that wikipedia can convey the (encyclopedic) information concerning them. Wikipedia is not a tabloid. Sapphorain (talk) 15:39, 14 February 2016 (UTC)

What Waring said

The present article does not seem to have a direct citation for Waring's own statement of his 'theorem'. Fortunately, Hilbert's paper of 1909 gives an exact source: Edward Waring, 'Meditationes Algebraicae', 3rd edn., 1782, p. 349-50. This is available at the Internet Archive. The cited pages contain several propositions, of which two (numbers 5 and 9) constitute what is now called Waring's Theorem (or Problem). The Latin text of these propositions is as follows:

5. Omnis integer numerus est quadratus; vel e duobus, tribus vel quatuor quadratis compositus.

9. Omnis integer numerus est cubus; vel e duobus, tribus, 4, 5, 6, 7, 8 vel novem cubis compositus: est etiam quadro-quadratus; vel e duobus, tribus, &c. usque ad novemdecim compositus, & sic deinceps: consimilia etiam affirmari possunt (exceptis excipiendis) de eodem numero quantitatum earundem dimensionum.

I have attempted a literal translation, with some help from Google Translate:

5. Every whole number is a square; or is composed of two, three or four squares.

9. Every whole number is a cube; or is composed of two, three, 4, 5, 6, 7, 8 or nine [novem] cubes: it is also a fourth power [literally a squared-square]; or is composed of two, three, etc. up to nineteen [novemdecim], and so on [& sic deinceps]: also similar things may be affirmed (with exceptions where exceptions are necessary) about the same number of quantities of the same dimensions.

The meaning of the last few words is not clear. I would take it as just a general assertion that similar statements can be made about higher powers. There is nothing to indicate that Waring actually had any formula for higher powers, let alone a proof. I don't know what he means about 'necessary exceptions'.

The Latin text incidentally rules out any possibility that '19' was a printing error for '16', not only because 16 would be incorrect, but because the number 'novemdecim' is printed as a word, not a numeral, and there is hardly a risk of anyone confusing it with the Latin for 'sixteen' ('sedecim').

I have not inserted a citation into the article, but someone else may wish to. I should also note that the text of 1782 is probably not Waring's first statement of his 'theorem', and there may be some variations in earlier editions of his book. I have not looked into this.2A00:23C8:7907:4B01:6448:1E21:4A4C:73C8 (talk) 13:58, 19 June 2022 (UTC)