Thursday, October 21, 2010

Waring's problem for cubes

Waring's problem for cubes : Every positive integer can be written as the sum of nine (or fewer) positive cubes.


6 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
7 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
8 = 2^3
9 = 2^3 + 1^3
10 = 2^3 + 1^3 + 1^3
11 = 2^3 + 1^3 + 1^3 + 1^3
12 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3
13 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
14 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
15 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
23 = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3.
24 = 2^3 + 2^3 + 2^3
29 = 3^3 + 1^3 + 1^3
31 = 3^3 + 1^3 + 1^3 + 1^3 + 1^3
51 = 3^3 + 2^3 + 2^3 + 2^3

Based on this, is it reasonable to suppose that every integer is the sum of nine positive cubes (or fewer) ?

23 is written as the sum of nine positive cubes. Can you find other numbers which can be written as the sum of nine positive cubes?




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