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. Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... 1770 was a common year starting on Monday (see link for calendar). ... Edward Waring (1736 - August 15, 1798) was British mathematician who was born in Old Heath (near Shrewsbury) Shropshire England and died in Pontesbury Shropshire England He was Lucasian professor of mathematics at Cambridge University from 1760 until his death. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... David Hilbert David Hilbert ( January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... 1909 was a common year starting on Friday (see link for calendar). ...

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 conjectured that these values were in fact the best possible.

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 g(2) = 4. Lagrange's four-square theorem was conjectured by Fermat in 1640 and was first stated in 1621. Lagranges four-square theorem, also known as Bachets Conjecture, was proved in 1770 by Joseph Louis Lagrange. ... 1770 was a common year starting on Monday (see link for calendar). ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer of Basque origin at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... Events December 1 - Portugal regains its independence from Spain and João IV of Portugal becomes king. ... Events February 9 - Gregory XV is elected pope. ...

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. Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... G. H. Hardy Godfrey Harold Hardy (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ... John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ...

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 Jing-run Chen and g(6) = 73 in 1940 by Pillai. 1909 was a common year starting on Friday (see link for calendar). ... 1912 is a leap year starting on Monday. ... Arthur Josef Alwin Wieferich (April 27, 1884 - September 15, 1954) was a German mathematician and teacher, remembered for his work on number theory. ... 1986 is a common year starting on Wednesday of the Gregorian calendar. ... 1964 was a leap year starting on Wednesday (link will take you to calendar). ... 1940 was a leap year starting on Monday (link will take you to calendar). ... Subbayya Sivasankaranarayana Pillai (1901-1950) was an Indian mathematician, well known for his work in number theory. ...

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) = floor((3/2)^k) + 2^k - 2     for k ≥ 6.

giving the values In 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. ...

1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190,132055 ... 1 (one) is the natural number following 0 and preceding 2. ... 4 (four) is a number, numeral, and glyph. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... 19 (nineteen) is the natural number following 18 and preceding 20. ... 37 is the natural number following 36 and preceding 38. ... 73 is the natural number following 72 and preceding 74. ...

listed in Sloane's A002804 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002804). Neil James Alexander Sloane is a US-American mathematician. ...

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) (ISBN 0-691-02351-4). Has a proof of the Lagrange theorem, accessible to high school students.