Problem of Waring

In Theory of the numbers, the problem of Waring , proposed in 1770 by Edward Waring, request if, for all Whole naturalness K , there exists a natural entirety S such as entire is the sum of with the more S powers K ^ième of entireties. The affirmative response was brought by David Hilbert in 1909. Sometimes, this subject is described like the theorem of Hilbert-Waring.

For each K , we note smallest S by G ( K ). We have G (1) = 1. Some simple calculations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 power fourth. Waring conjectured that these values were in fact the best possible ones.

The Théorème of the four squares of Lagrange of 1770 affirms that each natural number is the nap of with more the four squares; since three squares are not sufficient, this theorem establishes G (2) = 4. The theorem of the four squares of Lagrange was conjectured by Fermat in 1640 and its first mention goes back to 1621.

With the passing of years, various results on the values of G were established, by using increasingly sophisticated and complex techniques of demonstration. For example, Liouville showed that G (4) is worth to the more 53. Hardy and Littlewood showed that all the sufficiently large numbers are the sum of with more the 19 powers fourth.

The equality G (3) = 9 was established between 1909 and 1912 by Wieferich and A.J. Kempner, the equality G (4) = 19 in 1986 by R. Balasubramanian, F. Dress, and J. - M. Deshouillers, the equality G (5) = 37 in 1964 by Jing-run Chen and the equality G (6) = 73 in 1940 by Pillai.

All the other values of G are known today, thanks to the work of Dickson, Pillai, Rubugunday and Niven. Their statement contains two cases and it is conjectured that the second case can never occur; in the first case, the formula is read

G ( K ) = E ((3/2) ^K) + 2^K - 2 for K ≥ 6.

Readings to go further

W.J. Ellison: Waring' S problem . American Mathematical Monthly, volume 78 (1971), pp. 10-76. Exposed, containing a precise formula for G ( K ) and a simplified version of the proof of Hilbert.
Hans Rademacher and Otto Toeplitz, The Enjoyment off Mathematics (1933) (ISBN 0-691-02351-4). Of which a proof of the theorem of Lagrange, accessible to the students.

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