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Home > Waring's problem

In number theory, Waring's Problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s k^th powers of natural numbers. The affirmative answer was provided by David Hilbert in 1909. Sometimes this topic is described as Hilbert-Waring's theorem.

For every k, we denote the least such s by g(k). Note we have g(1) = 1. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth-powers. Waring conjected that these values were in fact the best possible.

Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that g(4) is at most 53. Hardy 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 19121912 is a leap year starting on Monday (link will take you to calendar) Events January 1 Establishment of Republic of China. January 6 New Mexico is admitted as the 47th U. January 17 British polar explorer Robert Falcon Scott and a team of four begin the by Wieferich and A. J. Kempner, g(4) = 19 in 19861986 is a common year starting on Wednesday. Events January January 1 Spain and Portugal enter the European Community January 1 Aruba gains increased autonomy from the Netherlands and is separated from the Netherlands Antilles. January 9 After losing a pa by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, g(5) = 37 in 1964Events January January 1 Federation of Rhodesia and Nyasaland is dissolved. January 3 Senator Barry Goldwater announces that he will seek the Republican nomination for President. January 5 In the first meeting between leaders of the Roman Catholic and Ort by Jing-run Chen and g(6) = 73 in 1940Events January-February January 5 FM radio is demonstrated to the FCC for the first time. January 6 World War II: Mass execution of Poles, committed by Germans in the Poznan, Warthegau. January 12 World War II: Russia bombs cities in Finland. February 2 F by PillaiSubbayya Sivasankaranarayana Pillai ( 1901- 1950) was an Indian mathematician, well known for his work in number theory. He was from Tamil Nadu. He worked on Waring's problem, in particular on the determination of exact values for the function g ''n ..

All the 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) = floorIn mathematics, the floor function is the function defined as follows: for a real number x floor x is the largest integer less than or equal to x''. For example, floor(2. 9) 2, floor(-2) -2 and floor(-2. The floor function is also denoted by x or. A more((3/2)^k) + 2^k - 2     for k ≥ 6.

Further Reading

• W. J. Ellison: Waring's problem. American Mathematical Monthly, volume 78 (1971), pp. 10-76. Survey, contains the precise formula for g(k) and a simplified version of Hilbert's proof.
• Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics (1933) (BooksEnthsiast.com). Has a proof of the Lagrange theorem, accessible to high school students.

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Topics: Waring's Problem Number Theorem Natural G''(''k Formula Squares...

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Waring's problemNumber theory Analytic number theory In number theory, Waring's Problem proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s ''k''th Warington Baden-PowellHenry Warington Smyth Baden-Powell Born 3 Feb 1847 Died 24 Apr 1921 at age 74. Known as Warington within the family, he was Robert Baden-Powell's oldest brother. He was educated at St Paul's College where he graduated in 1857. Early in his career he quali