Waring's problem

The Waring problem is a problem of number theory . It generalizes the four squares theorem , which says that any natural number can be represented as the sum of four square numbers. In his work Meditationes algebraicae ( 1770 ), Edward Waring suggested that there must be such a common number of summands for every exponent k . Waring's problem is now considered solved.

Waring's problem

Waring's generalization says that for every natural exponent there is a natural number , so that every natural number can be represented as the sum of -th powers. In addition, the smallest such number is then usually asked. For example, the four-squares theorem says that every natural number can be represented by a sum of four square numbers, that is . Since, as the number shows, three squares are not always enough, it has to be, so in total . ${\ displaystyle k}$ ${\ displaystyle g}$ ${\ displaystyle g}$ ${\ displaystyle k}$ ${\ displaystyle g (k)}$ ${\ displaystyle g (2) \ leq 4}$ ${\ displaystyle 7 = 2 ^ {2} + 1 ^ {2} + 1 ^ {2} + 1 ^ {2}}$ ${\ displaystyle g (2) \ geq 4}$ ${\ displaystyle g (2) = 4}$

So while you need 4 squares for the number 7, you need 9 cube numbers for the number 23 and 19 fourth powers for the number 79. Waring guessed that these values are the highest possible, so and . ${\ displaystyle g (2) = 4, g (3) = 9}$ ${\ displaystyle g (4) = 19}$

Solutions for small exponents

Waring's conjecture was proven by David Hilbert in 1909 . The statement is therefore sometimes referred to as the Waring-Hilbert theorem. Hilbert's proof was simplified in 1912 by Robert Remak and Erik Stridsberg . An elementary proof of Waring's problem, who used ideas other than Hilbert, was provided by Yuri Wladimirowitsch Linnik in 1942 using the results of Lew Schnirelman .

k = 2: is proved by the four-squares theorem . ${\ displaystyle g (2) = 4}$

k = 3: This is was proven in the years 1909 to 1912 by Arthur Wieferich and Aubrey J. Kempner (1880–1973). Edmund Landau was also able to show in 1909 that only a finite number of natural numbers require nine cubes, i.e. that each sufficiently large number can be represented as a sum of eight cubes, and Leonard E. Dickson found in 1939 that 23 and 239 are the only two numbers that actually require nine cubes. Arthur Wieferich already suspected that only 15 numbers actually need eight and only 121 numbers seven cubes. Today it is generally assumed that eight cubes are only required for numbers ≤ 454, seven cubes for numbers ≤ 8,042 and six cubes for numbers ≤ 1,290,740, i.e. all sufficiently large numbers can be represented as a sum of five cubes. Juri Linnik was the first to prove the seven-cube theorem in 1941; George Leo Watson simplified it significantly in 1951. ${\ displaystyle g (3) = 9}$

k = 4: was shown in 1986 by Ramachandran Balasubramanian , François Dress and Jean-Marc Deshouillers . It has also been known since 1939 that any sufficiently large number can be represented as a sum of 16 biquadrates , the set of numbers that actually require 17, 18 or 19 fourth powers, i.e. finite. This value cannot be improved. ${\ displaystyle g (4) = 19}$

k> 4: was detected in 1964 by Chen Jingrun . S. Sivasankaranarayana Pillai showed 1940 , and Leonard E. Dickson 1937 . ${\ displaystyle g (5) = 37}$ ${\ displaystyle g (6) = 73}$ ${\ displaystyle g (7) = 143}$

General solution

Through the work of Leonard Dickson, Pillai , RK Rubugunday and Ivan M. Niven , all other g (k) are now also known.

{\ displaystyle g (k) = {\ begin {cases} \ left \ lfloor \ left ({\ frac {3} {2}} \ right) ^ {k} \ right \ rfloor + 2 ^ {k} -2 & {\ text {for}} k \ leq 6 {\ text {or}} 3 ^ {k} -2 ^ {k} +2 <(2 ^ {k} -1) \ left \ lfloor \ left ({\ frac {3} {2}} \ right) ^ {k} \ right \ rfloor \\\ left \ lfloor \ left ({\ frac {3} {2}} \ right) ^ {k} \ right \ rfloor + \ left \ lfloor \ left ({\ frac {4} {3}} \ right) ^ {k} \ right \ rfloor + 2 ^ {k} -c_ {k} & {\ text {for}} k> 6 {\ text {and}} 3 ^ {k} -2 ^ {k} +2 \ geq (2 ^ {k} -1) \ left \ lfloor \ left ({\ frac {3} {2}} \ right ) ^ {k} \ right \ rfloor \ end {cases}}}

,

where and . ${\ displaystyle c_ {k} \ in \ {2,3 \}}$ ${\ displaystyle \ textstyle c_ {k} = 2 \ Leftrightarrow \ left \ lfloor \ left ({\ frac {3} {2}} \ right) ^ {k} \ right \ rfloor \ cdot \ left \ lfloor \ left ( {\ frac {4} {3}} \ right) ^ {k} \ right \ rfloor + \ left \ lfloor \ left ({\ frac {3} {2}} \ right) ^ {k} \ right \ rfloor + \ left \ lfloor \ left ({\ frac {4} {3}} \ right) ^ {k} \ right \ rfloor = 2 ^ {k}}$

It is assumed that the second case does not occur for any k . The condition for the first case is fulfilled for all and it is known that at most there can be a finite number for which the second case would even be considered. If this assumption is confirmed, the above formula could be used ${\ displaystyle 6 \ leq k \ leq 200,000}$ ${\ displaystyle k}$

{\ displaystyle g (k) = \ left \ lfloor \ left ({\ frac {3} {2}} \ right) ^ {k} \ right \ rfloor + 2 ^ {k} -2}

simplify.

k    1 2 3  4  5  6   7   8   9   10 …
g(k) 1 4 9 19 37 73 143 279 548 1079 …

(Follow A002804 in OEIS )

For larger ones , the number can also be estimated. ${\ displaystyle k}$ ${\ displaystyle g (k) \ approx 2 ^ {k}}$

Smallest number a (k)

The smallest number that requires the maximum number of summands in Waring's problem is for small ones : ${\ displaystyle a (k),}$ ${\ displaystyle g (k)}$ ${\ displaystyle k}$

k    1 2  3  4   5   6    7    8     9    10 …
g(k) 1 4  9 19  37  73  143  279   548  1079 …
a(k) 1 7 23 79 223 703 2175 6399 19455 58367 …

(Follow A018886 in OEIS )

Example for : Accordingly, every number can be represented as a sum of 9 powers of three (cubes). 23 is the smallest number that cannot be represented as the sum of less than 9 cubes, it is . ${\ displaystyle k = 3}$ ${\ displaystyle 23 = 2 ^ {3} + 2 ^ {3} + 1 ^ {3} + 1 ^ {3} + 1 ^ {3} + 1 ^ {3} + 1 ^ {3} + 1 ^ { 3} + 1 ^ {3}}$

Sources and literature

Edward Waring: Meditationes algebraicae. Cambridge ³ 1782.
Dennis Weeks (Ed.): Meditationes algebraicae. An English translation of the work of Edward Waring. Providence: American Mathematical Society, 1991. ISBN 0821801694 .

Web links

Wikisource: David Hilbert: Gesammelte Abhandlungen, Volume 1, 11. Proof for the representability of the whole numbers by a fixed number of n-th powers (Waring's problem). - Sources and full texts

Eric W. Weisstein : Waring's Problem . In: MathWorld (English).

Individual evidence

↑ David Hilbert: Proof of the representability of the whole numbers by a fixed number of -th powers (Waring's problem). ${\ displaystyle n}$ In: Mathematical Annals. 67 (1909), pp. 281-300. Cf. Erhard Schmidt: On Hilbert's proofs of Waring's theorem. (From a letter addressed to Mr. Hilbert.) In: Mathematische Annalen. 74 (1913), No. 2, pp. 271-274.

^ Erik Stridsberg: Sur la demonstration de M. (onsieur) Hilbert du théorème de Waring. In: Mathematical Annals. 72 (1912), pp. 145-152; Robert Remak: Comment on Mr. Stridsberg's proof of Waring's theorem. In: Mathematische Annalen 72 (1912), pp. 153–156.

↑ Yuri Vladimirovich Linnik: Элементарное решение проблемы Waring'a по методу Шнирельмана [= Elementarnoe rešenie problemy Waring'a po metodu Šnirel'mana. (Elementary solution of Waring's problem with Schnirelman's method.) ] In: Recueil Mathématique. Математический Сборник [= Matematičeskij Sbornik (Mathematical Collection)] NF 12/54 (1943), No. 2, pp. 225–230.

↑ Arthur Wieferich: Proof of the theorem that any whole number can be represented as a sum of at most nine positive cubes. In: Mathematical Annals. 66: 95-101 (1909).

^ Aubrey John Kempner: Comments on Waring's problem. In: Mathematical Annals. 72 (1912), pp. 387-399.

↑ Edmund Landau: About an application of prime number theory to Waring's problem in elementary number theory. In: Mathematical Annals. 66 (1909), pp. 102-105.

^ Leonard Eugene Dickson: All integers except 23 and 239 are sums of eight cubes. (PDF; 376 kB) In: Bulletin of the American Mathematical Society. 45, pp. 588-591 (1939).

↑ Arthur Wieferich: Proof of the theorem that any whole number can be represented as a sum of at most nine positive cubes. In: Mathematical Annals. 66 (1909), here p. 95: “Tables of the smallest numbers of positive cubes into which the whole numbers can be broken down have [...] been drawn up for the numbers up to 40,000. They showed that up to the limit of 40,000 all numbers greater than 239 can be represented by a maximum of 8, above 454 by a maximum of 7 and above 8,042 by a maximum of 6 cubes, so that presumably beyond a certain limit (8,042) each whole Number can be represented as a sum of a maximum of 6 cubes. "

↑ See WS Baer: On the division of whole numbers into seven cubes. In: Mathematical Annals. 74 (1913), No. 4, pp. 511-514. - François Bertault; Olivier Ramare; Paul Zimmermann: On sums of seven cubes. (PDF; 254 kB) In: Mathematics of Computation. 68 (1999), No. 227, pp. 1303-1310.

^ Juri Wladimirowitsch Linnik: On the representation of large numbers as sums of seven cubes. ( О разложении больших чисел на семь кубов [= O razloženii bol'šich čsel na sem 'kubov. (About the representation of large numbers as the sum of seven cubes.) ]) In: Comptes Rendus (Docady) de l'Académie de l'Académie l'URSS. NF 35 (1942), No. 6, p. 162 ff. Also in: Recueil Mathématique Математический Сборник [= Matematičeskij Sbornik (Mathematical Collection)] NF 12/54 (1943), No. 2, pp. 218-224.

↑ George Leo Watson: A proof of the seven cubes theorem. In: Journal of the London Mathematical Society. 26: 153-156 (1951).

↑ Ramachandran Balasubramanian; Jean-Marc Deshouillers; François Dress: Problems of Waring pour les bicarrés. In: Comptes rendus de l'Académie des sciences. Series I: Mathematique 303 (1986), No. 4, pp. 85-88, No. 5, pp. 161-163.

↑ Harold Davenport: On Waring's Problem for Fourth Powers. In: Annals of Mathematics. 40 (1939), No. 4, pp. 731-747.

^ Aubrey John Kempner: Comments on Waring's problem. In: Mathematical Annals. 72 (1912), here pp. 395–396 (§ 4. Numbers that require at least 16 positive biquadrates occur again and again in the natural number series).

↑ Chen Jingrun: Waring's Problem for g (5) = 37. ( Memento of the original from December 18, 2015 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 501 kB) In: Scientia Sinica. 13 (1964), pp. 1547-1568. Also in: Chinese Mathematics. Acta Scientiarum Mathematicarum. 6, pp. 105-127 (1965). @1@ 2

↑ Subbayya Sivasankaranarayana Pillai: On Waring's problem-g (6) = 73. In: Proceedings of the Indian Academy of Sciences A 12 (1940). Pp. 30-40.

^ Leonard Eugene Dickson: The Waring Problem and its generalizations. In: Bulletin of the American Mathematical Society. 42 (1936), pp. 833-842.

↑ Subbayya Sivasankaranarayana Pillai: On Waring's Problem. In: Journal of the Indian Mathematical Society. 2 (1936), No. 2, pp. 16-44; see. Sarvadaman Chowla: Pillai's Exact Formulas for the Number g (n) in Waring's Problem. In: Proceedings of the Indian Academy of Sciences. A 4 (1936), p. 261.

↑ Shri Raghunath Krishna Rubugunday: On g (k) in Waring's problem. In: Journal of the Indian Mathematical Society. 6 (1942), No. 2, pp. 192-198.

↑ Ivan Morton Niven: An unsolved case of the Waring Problem. In: American Journal of Mathematics. 66, pp. 137-143 (1944).

^ WJ Ellison: Waring's problem. In: American Mathematical Monthly. 1971, Volume 78, pages 10-36, Theorem 4.1.

↑ RM Stemmler: The ideal Waring theorem for exponents 401-200,000. In: Math. Comp. 1964, Volume 18, Pages 144-146.

↑ K. Mahler: On the fractional parts of powers of real numbers. In: Mathematika. 1957, volume 4, pages 122-124.

[1] David Hilbert: Proof of the representability of the whole numbers by a fixed number of -th powers (Waring's problem). ${\ displaystyle n}$ In: Mathematical Annals. 67 (1909), pp. 281-300. Cf. Erhard Schmidt: On Hilbert's proofs of Waring's theorem. (From a letter addressed to Mr. Hilbert.) In: Mathematische Annalen. 74 (1913), No. 2, pp. 271-274.

[2] Erik Stridsberg: Sur la demonstration de M. (onsieur) Hilbert du théorème de Waring. In: Mathematical Annals. 72 (1912), pp. 145-152; Robert Remak: Comment on Mr. Stridsberg's proof of Waring's theorem. In: Mathematische Annalen 72 (1912), pp. 153–156.

[3] Yuri Vladimirovich Linnik: Элементарное решение проблемы Waring'a по методу Шнирельмана [= Elementarnoe rešenie problemy Waring'a po metodu Šnirel'mana. (Elementary solution of Waring's problem with Schnirelman's method.) ] In: Recueil Mathématique. Математический Сборник [= Matematičeskij Sbornik (Mathematical Collection)] NF 12/54 (1943), No. 2, pp. 225–230.

[4] Arthur Wieferich: Proof of the theorem that any whole number can be represented as a sum of at most nine positive cubes. In: Mathematical Annals. 66: 95-101 (1909).

[5] Aubrey John Kempner: Comments on Waring's problem. In: Mathematical Annals. 72 (1912), pp. 387-399.

[6] Edmund Landau: About an application of prime number theory to Waring's problem in elementary number theory. In: Mathematical Annals. 66 (1909), pp. 102-105.

[7] Leonard Eugene Dickson: All integers except 23 and 239 are sums of eight cubes. (PDF; 376 kB) In: Bulletin of the American Mathematical Society. 45, pp. 588-591 (1939).

[8] Arthur Wieferich: Proof of the theorem that any whole number can be represented as a sum of at most nine positive cubes. In: Mathematical Annals. 66 (1909), here p. 95: “Tables of the smallest numbers of positive cubes into which the whole numbers can be broken down have [...] been drawn up for the numbers up to 40,000. They showed that up to the limit of 40,000 all numbers greater than 239 can be represented by a maximum of 8, above 454 by a maximum of 7 and above 8,042 by a maximum of 6 cubes, so that presumably beyond a certain limit (8,042) each whole Number can be represented as a sum of a maximum of 6 cubes. "

[9] See WS Baer: On the division of whole numbers into seven cubes. In: Mathematical Annals. 74 (1913), No. 4, pp. 511-514. - François Bertault; Olivier Ramare; Paul Zimmermann: On sums of seven cubes. (PDF; 254 kB) In: Mathematics of Computation. 68 (1999), No. 227, pp. 1303-1310.

[10] Juri Wladimirowitsch Linnik: On the representation of large numbers as sums of seven cubes. ( О разложении больших чисел на семь кубов [= O razloženii bol'šich čsel na sem 'kubov. (About the representation of large numbers as the sum of seven cubes.) ]) In: Comptes Rendus (Docady) de l'Académie de l'Académie l'URSS. NF 35 (1942), No. 6, p. 162 ff. Also in: Recueil Mathématique Математический Сборник [= Matematičeskij Sbornik (Mathematical Collection)] NF 12/54 (1943), No. 2, pp. 218-224.

[11] George Leo Watson: A proof of the seven cubes theorem. In: Journal of the London Mathematical Society. 26: 153-156 (1951).

[12] Ramachandran Balasubramanian; Jean-Marc Deshouillers; François Dress: Problems of Waring pour les bicarrés. In: Comptes rendus de l'Académie des sciences. Series I: Mathematique 303 (1986), No. 4, pp. 85-88, No. 5, pp. 161-163.

[13] Harold Davenport: On Waring's Problem for Fourth Powers. In: Annals of Mathematics. 40 (1939), No. 4, pp. 731-747.

[14] Aubrey John Kempner: Comments on Waring's problem. In: Mathematical Annals. 72 (1912), here pp. 395–396 (§ 4. Numbers that require at least 16 positive biquadrates occur again and again in the natural number series).