Waring's problem explained

In number theory, Waring's problem asks whether each natural number k has an associated positive integer 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 in 1909.^[1] Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".

Relationship with Lagrange's four-square theorem

Long before Waring posed his problem, Diophantus 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, and it was solved by Joseph-Louis Lagrange in his four-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

, let denote the minimum number of th powers of naturals needed to represent all positive integers. Every positive integer is the sum of one first power, itself, so . Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes,^[2] and 79 requires 19 fourth powers; these examples show that , , and . Waring conjectured that these lower bounds were in fact exact values.^[3]

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

. Lagrange's four-square theorem was conjectured in Bachet's 1621 edition of Diophantus's Arithmetica; Fermat claimed to have a proof, but did not publish it.^[4]

Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that

is at most 53. Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers.

Exact values for g(k)
Value	Year of discovery	Author
g(2) = 4	1770	J.-L. Lagrange[5]
g(3) = 9	1909	A. Wieferich[6] A gap in the proof was filled by A. J. Kempner in 1912[7]
g(4) = 19	1986	R. Balasubramanian, J.-M. Deshouillers and F. Dress[8] [9]
g(5) = 37	1964	J. R. Chen[10] and J. H. Conway[11] (independently)
g(6) = 73	1940	S. S. Pillai[12]
g(7) = 143	1936	L. E. Dickson[13] and S. S. Pillai[14] (independently)
g(k), k > 7	1936–1944	L. E. Dickson[15] and S. S. Pillai[16] [17] (independently) in 1936, for almost all cases, the rest of which were treated by R. K. Rubugunday in 1942[18] and I. M. Niven in 1944[19]

Let and be defined by the Euclidean divisionor explicitly by and , where and respectively denote the integral and fractional part of a real number .

Since the number

is less than , as a sum of integer powers, it can only be expressed using and . Using modular arithmetic, one shows that the fewest number of terms is achieved by the formula$$2^k\; q\; -\; 1\; =\; \backslash underbrace\_\; +\; \backslash underbrace\_,$$and it follows that$$g(k)\; \backslash ge\; 2^k\; +\; q\; -\; 2,$$which was noted by J. A. Euler in about 1772.^[20] Let . Combined work from the authors quoted above has led to the formula,^{[21] [22]} valid for all :Dickson and Pillai both independently proved the first case, for , and the two other cases,^[23] and they noted that for . Rubugunday proved that for all , leaving the final case open. In this scenario, Niven proved that .

No value of

is known for which the hypothesis in the last two cases holds. Mahler^[24] proved that there can only be a finite number of such . Kubina and Wunderlich^[25], extending work of Stemmler^[26], have shown that any such must satisfy . It is conjectured that there are no such ; in that case, for every positive integer .

The first few values of

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 Hardy and Littlewood,^[27] 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) (and also to 0, 1, or 4 (mod 5)^[28]), 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 showed^[29] 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 1986^[30] and 1989^[31] reduced the 14 biquadrates 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)

Bounds
1 = G(1) = 1
4 = G(2) = 4
4 ≤ G(3) ≤ 7
16 = G(4) = 16
6 ≤ G(5) ≤ 17
9 ≤ G(6) ≤ 24
8 ≤ G(7) ≤ 33
32 ≤ G(8) ≤ 42
13 ≤ G(9) ≤ 50
12 ≤ G(10) ≤ 59
12 ≤ G(11) ≤ 67
16 ≤ G(12) ≤ 76
14 ≤ G(13) ≤ 84
15 ≤ G(14) ≤ 92
16 ≤ G(15) ≤ 100
64 ≤ G(16) ≤ 109
18 ≤ G(17) ≤ 117
27 ≤ G(18) ≤ 125
20 ≤ G(19) ≤ 134
25 ≤ G(20) ≤ 142

The number G(k) is greater than or equal to

2r+2	if k = 2r with r ≥ 2, or k = 3 × 2r;
pr+1	if p is a prime greater than 2 and k = pr(p - 1);
(pr+1 − 1)/2	if p is a prime greater than 2 and k = pr(p - 1)/2;
k + 1	for all integers k greater than 1.

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 2N requiring 5 cubes drops off with increasing N at sufficient speed to have people believe that ;^[32] the largest number now known not to be a sum of 4 cubes is,^[33] and the authors give reasonable arguments there that this may be the largest possible. The upper bound is due to Linnik in 1943.^[34] (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^[35] that every number between and 10²⁴⁵ required at most 16, and Kawada, Wooley and Deshouillers extended^[36] Davenport's 1939 result to show that every number above 10²²⁰ required at most 16). Numbers of the form 31·16ⁿ always require 16 fourth powers.

is the last known number that requires 9 fifth powers (Integer sequence S001057, Tony D. Noe, Jul 04 2017), 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 and Wooley.^[37]

Using his improved Hardy–Ramanujan–Littlewood method, I. M. Vinogradov published numerous refinements leading to

in 1947^[38] and, ultimately, for an unspecified constant C and sufficiently large k in 1959.^[39]

Applying his p-adic form of the Hardy–Ramanujan–Littlewood–Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Anatolii Alexeevitch Karatsuba obtained^[40] in 1985 a new estimate, for

:

Further refinements were obtained by Vaughan in 1989.^[41]

Wooley then established that for some constant C,^[42]

Vaughan and Wooley's survey article from 2002 was comprehensive at the time.^[37]

See also

• Centered polygonal number theorem
• Fermat polygonal number theorem, that every positive integer is a sum of at most n of the n-gonal numbers
• Jacobi's four-square theorem, provides the number of ways a positive integer can be represented as the sum of 4 squares
• Pollock's conjectures
• Subset sum problem, an algorithmic problem that can be used to find the shortest representation of a given number as a sum of powers
• Sums of four cubes problem, discusses whether every integer is the sum of four cubes of integers
• Sums of three cubes, discusses what numbers are the sum of three not necessarily positive cubes
• Waring–Goldbach problem, the problem of representing numbers as sums of powers of primes

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, "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).
• Ellison . W. J. . 1971 . Waring's problem . American Mathematical Monthly . 78 . 1. 10–36 . 10.2307/2317482. 2317482 . Survey, contains the precise formula for G(k), a simplified version of Hilbert's proof and a wealth of references.
• Book: Khinchin, A. Ya. . Aleksandr Khinchin . Three Pearls of Number Theory . Dover . Mineola, NY . 1998 . 978-0-486-40026-6 . Has an elementary proof of the existence of G(k) using Schnirelmann density.
• Book: Nathanson, Melvyn B. . Additive Number Theory: The Classical Bases . 164 . . . 1996 . 0-387-94656-X . 0859.11002 . Has proofs of Lagrange's theorem, the polygonal number theorem, 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 kth powers.
• Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics (1933) . Has a proof of the Lagrange theorem, accessible to high-school students.

Notes and References

1. David . Hilbert . David Hilbert . Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem) . de . . 67 . 281–300 . 1909 . 3 . 10.1007/bf01450405 . 1511530. 179177986 .
2. Remember we restrict ourselves to positive natural numbers. With general integers, it is not hard to write 23 as the sum of 4 cubes, e.g.

   2³+2³+2³+(-1)³

   or

   29³+17³+8³+(-31)³

   .
3. Ellison . W. J. . 1971 . Waring's Problem . The American Mathematical Monthly . 78 . 1 . 10 . 10.2307/2317482.
4. Book: Dickson, Leonard Eugene . Leonard Eugene Dickson . History of the Theory of Numbers . II: Diophantine Analysis . . 1920 . Chapter VIII.
5. Lagrange . Joseph-Louis . 1770 . Démonstration d'un théorème d'arithmétique . Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin . 123-133.
6. Wieferich . Arthur . Arthur Wieferich . 1909 . Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt . Mathematische Annalen . de . 66 . 1 . 95–101 . 10.1007/BF01450913 . 121386035.
7. Kempner . Aubrey . 1912 . Bemerkungen zum Waringschen Problem . Mathematische Annalen . de . 72 . 3 . 387–399 . 10.1007/BF01456723 . 120101223.
8. Balasubramanian . Ramachandran . Deshouillers . Jean-Marc . Dress . François . 1986 . Problème de Waring pour les bicarrés. I. Schéma de la solution . Waring's problem for biquadrates. I. Sketch of the solution . Comptes Rendus de l'Académie des Sciences, Série I . fr . 303 . 4 . 85–88 . 0853592.
9. Balasubramanian . Ramachandran . Deshouillers . Jean-Marc . Dress . François . 1986 . Problème de Waring pour les bicarrés. II. Résultats auxiliaires pour le théorème asymptotique . Waring's problem for biquadrates. II. Auxiliary results for the asymptotic theorem . Comptes Rendus de l'Académie des Sciences, Série I . fr . 303 . 5 . 161–163 . 0854724.
10. Chen . Jing-Run . 1964 . Waring's problem for g(5)=37 . Scientia Sinica . en . 13 . 10 . 1547–1568 .
11. Conway originally intended to publish this result as part of his PhD thesis. However, Chen published the same result later that year, leading Conway to forgo publication and instead change the topic of his thesis. See
    . September 2005 . Breakfast with John Horton Conway . EMS Newsletter . 57 . 32–34.
    Web site: 2022 . John Horton Conway (December 26, 1937 – April 11, 2020) . 2026-02-13 . MacTutor . PDF.
12. Pillai . S. S. . 1940 . On Waring's problem g(6) = 73 . Proc. Indian Acad. Sci. . 12 . 30–40 . 10.1007/BF03170721 . 0002993 . 185097940.
13. Dickson . L. E. . 1936 . Proof of the Ideal Waring Theorem for Exponents 7-180 . American Journal of Mathematics . 58 . 3 . 521–529 . 10.2307/2370969 . 0002-9327. subscription .
14. Pillai . S. S. . 1936 . On Waring's Problem IV . Journal Of The Annamalai University . VI . 54-64.
15. Dickson . L. E. . 1936 . Solution of Waring's Problem . American Journal of Mathematics . 58 . 3 . 530–535 . 10.2307/2370970 . 2370970.
16. Pillai . S.S. . 1936 . On Waring's Problem . Journal of the Indian Mathematical Society . 2 . 16–44.
17. Pillai . S. S. . 1936 . On Waring's Problem III . Journal Of The Annamalai University . VI . 50-53.
18. Rubugunday . R. K. . 1942 . On g(k) in Waring's Problem . Journal of the Indian Mathematical Society . 6 . 192–198.
19. Niven . Ivan M. . Ivan M. Niven . 1944 . An unsolved case of the Waring problem . . The Johns Hopkins University Press . 66 . 1 . 137–143 . 10.2307/2371901 . 2371901 . 0009386.
20. Book: Euler, Leonhard . Leonhardi Euleri Opera postuma mathematica et physica : anno MDCCCXLIV detecta . 1862 . Petropoli : Eggers . Harvard University . 203–204.
21. Book: Hardy, Godfrey H. . An introduction to the theory of numbers . Wright . Edward M. . Heath-Brown . D. R. . 2008 . Oxford University Press . 978-0-19-921986-5 . 6. ed., [rev. by D. R. Heath-Brown ...] . Oxford mathematics . Oxford.
22. Small . Charles . 1977-01-01 . Waring's Problem . Mathematics Magazine . en . 50 . 1 . 12 . 10.2307/2689743.
23. For the secondary condition, Pillai gives

    r+s\ge3^k

    and

    r+s<3^k

    instead of Dickson's

    qd+q+d=2^k

    and

    qd+q+d>2^k

    , which are equivalent under

    q+r>2^k

    .
24. Mahler . Kurt . 1957 . On the fractional parts of the powers of a rational number II . . 4 . 2 . 122–124 . 10.1112/s0025579300001170 . 0093509.
25. Kubina . Jeffrey M. . Wunderlich . Marvin C. . 1990 . Extending Waring's conjecture to 471,600,000 . . 55 . 192 . 815–820 . 1990MaCom..55..815K . 10.2307/2008448 . 2008448 . 1035936.
26. Stemmler . Rosemarie M. . 1964 . The ideal Waring theorem for exponents 401-200,000 . Mathematics of Computation . 18 . 85 . 144–146 . 10.1090/S0025-5718-1964-0159803-X . 0025-5718 . 4 February 2025 . free.
27. Hardy . G. H. . Littlewood . J. E. . Some problems of Partitio Numerorum: IV. The singular series in Waring's Problem and the value of the number G(k) . Mathematische Zeitschrift . 12 . 1 . 1922 . 0025-5874 . 10.1007/BF01482074 . 161–188.
28. Web site: Square Modulo 5 .
29. H. . Davenport . Harold Davenport . On Waring's Problem for Fourth Powers . en . . 40 . 731–747 . 1939 . 4 . 10.2307/1968889 . 1968889 . 1939AnMat..40..731D .
30. Vaughan . R. C. . On Waring's Problem for Smaller Exponents . Proceedings of the London Mathematical Society . s3-52 . 3 . 1986 . 10.1112/plms/s3-52.3.445 . 445–463.
31. Vaughan . R. C. . A new iterative method in Waring's problem . Acta Mathematica . 162 . 1989 . 0001-5962 . 10.1007/BF02392834 . 1–71.
32. .
33. Deshouillers . Jean-Marc . Hennecart . François . Landreau . Bernard . I. Gusti Putu Purnaba . Appendix by . 7373170279850 . Mathematics of Computation . 69 . 229 . 2000 . 421–439 . 10.1090/S0025-5718-99-01116-3 . free.
34. U. V. Linnik. "On the representation of large numbers as sums of seven cubes". Mat. Sb. N.S. 12(54), 218–224 (1943).
35. Deshouillers . Jean-Marc . Hennecart . François . Landreau . Bernard . Waring's Problem for sixteen biquadrates – numerical results . . 12 . 2 . 2000 . 411–422 . 10.5802/jtnb.287 . free. subscription .
36. Deshouillers . Jean-Marc . Kawada . Koichi . Wooley . Trevor D. . On Sums of Sixteen Biquadrates . Mémoires de la Société Mathématique de France . 1 . 2005 . 0249-633X . 10.24033/msmf.413 . 1–120.
37. Book: R. C. . Vaughan . Trevor Wooley . Trevor . Wooley . Waring's Problem: A Survey . Number Theory for the Millennium . III . A. K. Peters . 301–340 . 2002 . 978-1-56881-152-9 . 1956283 . Natick, MA . Bennet . Michael A. . Berndt . Bruce C. . Boston . Nigel . Diamond . Harold G. . Hildebrand . Adolf J. . Walter . Philipp.
38. Book: Vinogradov, Ivan Matveevich . Roth . K.F. . Davenport . Anne . The Method of Trigonometrical Sums in the Theory of Numbers . Dover Publications . Mineola, NY . 1 Sep 2004 . 1947 . 978-0-486-43878-8.
39. Vinogradov . I. M. . On an upper bound for $G(n)$ . Izv. Akad. Nauk SSSR Ser. Mat. . 1959 . 23 . 5 . 637–642 . Russian.
40. A. A. . Karatsuba . On the function G(n) in Waring's problem . Izv. Akad. Nauk SSSR Ser. Mat. . 27 . 4 . 935–947 . 1985 . 1986IzMat..27..239K . 10.1070/IM1986v027n02ABEH001176.
41. Vaughan . R. C. . A new iterative method in Waring's problem . Acta Mathematica . 162 . 1989 . 0001-5962 . 10.1007/BF02392834 . 1–71.
42. Book: Vaughan, R. C. . 0868.11046 . The Hardy–Littlewood method . 2nd . Cambridge Tracts in Mathematics . 125 . Cambridge . . 1997 . 0-521-57347-5 .