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1. Waring's problemby zhixuanli pure mathematics number theory

2. proposed in 1770 by Edward Waring in his MeditationesAlgebraice • “Every integer is a cube or the sum of two, three, … nine cubes; every integer is also the square of a square, or the sum of up to nineteen such; and so forth."

3. examples • 2 = 12 + 12 + 02 + 02 • 4 = 12 + 12 + 12 + 12 • 9 = 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 • 15 = 32 + 22 + 12 + 12 • 15 = 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 03

4. Generalization • For every natural number k there exists an associated positive integer s such that every natural number is the sum of no more than a fixed s of positive kth integer powers

5. For every k, we denote by g(k) the minimum number s of kth powers needed to represent all integers g(k) = s • What is the relation between k and s ?

6. g(2) = 4 was proved by Lagrange's four-square theorem in 1770 • g(3) = 9 was established from 1909 to 1912 by Wieferichand A. J. Kempner • g(4) = 19 was proved in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers • g(5) = 37 was proved in 1964 by Chen Jingrun • g(6) = 73 was proved in 1940 by Pillai

7. In 1772, J. A. Euler conjectured that • has been verified to be correct for 6 ≦ k ≦ 471600000

8. Extended problem • There is also a related problem of finding the least integer n such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(k) number of kth powers of positive integers.

9. G(2) = 4 • G(4) = 16 , proved by Davenport in 1939

10. referernce • http://www.math.uga.edu/~pollack/riegerdress.pdf • http://www.personal.psu.edu/rcv4/Waring.pdf • http://mathworld.wolfram.com/WaringsProblem.html • http://www.ams.org/journals/mcom/1990-54-190/S0025-5718-1990-1011440-6/S0025-5718-1990-1011440-6.pdf