number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

, Waring's problem asks whether each

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

''k'' has an associated

positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

in 1909. Waring's problem has its own

Mathematics Subject Classification The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH ...

, 11P05, "Waring's problem and variants".

Relationship with Lagrange's four-square theorem

Long before Waring posed his problem,

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

had asked whether every positive integer could be represented as the sum of four perfect squares greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by

Claude Gaspard Bachet de Méziriac Claude may refer to: __NOTOC__ People and fictional characters * Claude (given name), a list of people and fictional characters * Claude (surname), a list of people * Claude Lorrain (c. 1600–1682), French landscape painter, draughtsman and etcher ...

, and it was solved by

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiafour-square theorem in 1770, the same year Waring made his conjecture. Waring sought to generalize this problem by trying to represent all positive integers as the sum of cubes, integers to the fourth power, and so forth, to show that any positive integer may be represented as the sum of other integers raised to a specific exponent, and that there was always a maximum number of integers raised to a certain exponent required to represent all positive integers in this way.

The number ''g''(''k'')

For every

k

, let

g(k)

denote the minimum number

s

k

th powers of naturals needed to represent all positive integers. Every positive integer is the sum of one first power, itself, so

g(1) = 1

. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth powers; these examples show that

g(2) \ge 4

g(3) \ge 9

, and

g(4) \ge 19

. Waring conjectured that these lower bounds were in fact exact values.

Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four. p = a_0^2 + a_1^2 + a_2^2 + a_ ...

of 1770 states that every natural number is the sum of at most four squares. Since three squares are not enough, this theorem establishes

g(2) = 4

. Lagrange's four-square theorem was conjectured in Bachet's 1621 edition of

Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate ...

;

Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...

claimed to have a proof, but did not publish it. Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example,

Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...

showed that

g(4)

is at most 53.

Hardy Hardy may refer to: People * Hardy (surname) * Hardy (given name) * Hardy (singer), American singer-songwriter Places Antarctica * Mount Hardy, Enderby Land * Hardy Cove, Greenwich Island * Hardy Rocks, Biscoe Islands Australia * Hardy, Sout ...

and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers. That

g(3) = 9

was established from 1909 to 1912 by Wieferich and A. J. Kempner,

g(4) = 19

in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers,

g(5) = 37

in 1964 by

Chen Jingrun Chen Jingrun (; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime. Life and career Chen was the third son i ...

, and

g(6) = 73

in 1940 by Pillai. Let

\lfloor x\rfloor

and

\

respectively denote the

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

and

fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part ca ...

of a positive real number

x

. Given the number

c = 2^k \lfloor(3/2)^k\rfloor - 1 < 3^k

, only

2^k

and

1^k

can be used to represent

c

; the most economical representation requires

\lfloor(3/2)^k\rfloor - 1

terms of

2^k

and

2^k - 1

terms of

1^k

. It follows that

g(k)

is at least as large as

2^k + \lfloor(3/2)^k\rfloor - 2

. This was noted by J. A. Euler, the son of

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

, in about 1772. Later work by Dickson, Pillai, Rubugunday, Niven and many others has proved that :

g(k) = \begin
 2^k + \lfloor(3/2)^k\rfloor - 2
  &\text\quad
  2^k \ + \lfloor(3/2)^k\rfloor \le 2^k, \\
 2^k + \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor - 2
  &\text\quad
  2^k \ + \lfloor(3/2)^k\rfloor > 2^k
  \text
  \lfloor(4/3)^k\rfloor \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor + \lfloor(3/2)^k\rfloor = 2^k, \\
 2^k + \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor - 3
  &\text\quad
  2^k \ + \lfloor(3/2)^k\rfloor > 2^k
  \text
  \lfloor(4/3)^k\rfloor \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor + \lfloor(3/2)^k\rfloor > 2^k.
\end

No value of

k

is known for which

2^k\ + \lfloor(3/2)^k\rfloor > 2^k

Mahler Gustav Mahler (; 7 July 1860 – 18 May 1911) was an Austro-Bohemian Romantic composer, and one of the leading conductors of his generation. As a composer he acted as a bridge between the 19th-century Austro-German tradition and the modernism ...

proved that there can only be a finite number of such

k

, and Kubina and Wunderlich have shown that any such

k

must satisfy

k > 471\,600\,000

. Thus it is conjectured that this never happens, that is,

g(k) = 2^k + \lfloor(3/2)^k\rfloor - 2

for every positive integer

k

. The first few values of

g(k)

are: : 1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, ... .

The number ''G''(''k'')

From the work of

and Littlewood, the related quantity ''G''(''k'') was studied with ''g''(''k''). ''G''(''k'') is defined to be the least positive integer ''s'' such that every sufficiently large integer (i.e. every integer greater than some constant) can be represented as a sum of at most ''s'' positive integers to the power of ''k''. Clearly, ''G''(1) = 1. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that . Since for all ''k'', this shows that .

Davenport Davenport may refer to: Places Australia *Davenport, Northern Territory, a locality *Hundred of Davenport, cadastral unit in South Australia **Davenport, South Australia, suburb of Port Augusta ** District Council of Davenport, former local govern ...

showed that 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). The exact value of ''G''(''k'') is unknown for any other ''k'', but there exist bounds.

Lower bounds for ''G''(''k'')

The number ''G''(''k'') is greater than or equal to : In the absence of congruence restrictions, a density argument suggests that ''G''(''k'') should equal .

Upper bounds for ''G''(''k'')

''G''(3) is at least 4 (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3, is the last to require 6 cubes, and the number of numbers between ''N'' and 2''N'' requiring 5 cubes drops off with increasing ''N'' at sufficient speed to have people believe that ; the largest number now known not to be a sum of 4 cubes is , and the authors give reasonable arguments there that this may be the largest possible. The upper bound is due to Linnik in 1943. (All nonnegative integers require at most 9 cubes, and the largest integers requiring 9, 8, 7, 6 and 5 cubes are conjectured to be 239, 454, 8042, and , respectively.) is the largest number to require 17 fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 that every number between and 10²⁴⁵ required at most 16, and Kawada, Wooley and Deshouillers extended Davenport's 1939 result to show that every number above 10²²⁰ required no more than 16). Numbers of the form 31·16^''n'' always require 16 fourth powers. is the last number less than 1.3 that requires 10 fifth powers, and is the last number less than 1.3 that requires 11. The upper bounds on the right with are due to

Vaughan Vaughan () (2021 population 323,103) is a city in Ontario, Canada. It is located in the Regional Municipality of York, just north of Toronto. Vaughan was the fastest-growing municipality in Canada between 1996 and 2006 with its population increas ...

and Wooley. Using his improved Hardy-Littlewood method, I. M. Vinogradov published numerous refinements leading to :

G(k)\le k(3\log k + 11)

in 1947 and, ultimately, :

G(k) \le k(2\log k + 2\log\log k + C\log\log\log k)

for an unspecified constant ''C'' and sufficiently large ''k'' in 1959. Applying his ''p''-adic form of the Hardy–Littlewood–Ramanujan–Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors,

Anatolii Alexeevitch Karatsuba Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (russian: Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, 31 January 1937 – Moscow, Russia, 28 September 2008) was a Russian mathematician ...

obtained (1985) a new estimate of the

function

G(k)

(for

k \ge 400

): :

G(k) < 2 k\log k + 2 k\log\log k + 12 k.

Further refinements were obtained by Vaughan in 1989. Wooley then established that for some constant ''C'', :

G(k) \le k\log k + k\log\log k + Ck.

Vaughan and Wooley have written a comprehensive survey article.

Notes

References

* G. I. Arkhipov, V. N. Chubarikov, A. A. Karatsuba, "Trigonometric sums in number theory and analysis". Berlin–New-York: Walter de Gruyter, (2004). * G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, "Theory of multiple trigonometric sums". Moscow: Nauka, (1987). * Yu. V. Linnik, "An elementary solution of the problem of Waring by Schnirelman's method". ''Mat. Sb., N. Ser.'' 12 (54), 225–230 (1943). * R. C. Vaughan, "A new iterative method in Waring's problem". ''Acta Mathematica'' (162), 1–71 (1989). *

I. M. Vinogradov Ivan Matveevich Vinogradov ( rus, Ива́н Матве́евич Виногра́дов, p=ɪˈvan mɐtˈvʲejɪvʲɪtɕ vʲɪnɐˈɡradəf, a=Ru-Ivan_Matveyevich_Vinogradov.ogg; 14 September 1891 – 20 March 1983) was a USSR, Soviet mathemati ...

, "The method of trigonometrical sums in the theory of numbers". ''Trav. Inst. Math. Stekloff'' (23), 109 pp. (1947). * I. M. Vinogradov, "On an upper bound for ''G''(''n'')". ''Izv. Akad. Nauk SSSR Ser. Mat.'' (23), 637–642 (1959). * I. M. Vinogradov, A. A. Karatsuba, "The method of trigonometric sums in number theory", ''Proc. Steklov Inst. Math.'', 168, 3–30 (1986); translation from Trudy Mat. Inst. Steklova, 168, 4–30 (1984). * Survey, contains the precise formula for ''G''(''k''), a simplified version of Hilbert's proof and a wealth of references. * Has an elementary proof of the existence of ''G''(''k'') using

Schnirelmann density In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.Schnirelmann, L.G. (1930).On the ...

. * Has proofs of Lagrange's theorem, the

polygonal number theorem In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum ...

, Hilbert's proof of Waring's conjecture and the Hardy–Littlewood proof of the asymptotic formula for the number of ways to represent ''N'' as the sum of ''s'' ''k''th powers. *

Hans Rademacher Hans Adolph Rademacher (; 3 April 1892, Wandsbeck, now Hamburg-Wandsbek – 7 February 1969, Haverford, Pennsylvania, USA) was a German-born American mathematician, known for work in mathematical analysis and number theory. Biography Rademacher r ...

and

Otto Toeplitz Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in Life and work Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnas ...

, ''The Enjoyment of Mathematics'' (1933) (). Has a proof of the Lagrange theorem, accessible to high-school students.

External links

* {{DEFAULTSORT:Waring's Problem Additive number theory Mathematical problems Unsolved problems in number theory Squares in number theory