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Problem of the Week 1080

Sums of Cubes

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Express each integer between 1 and 1000 as a sum of four cubes of integers (which can be positive, negative, or zero).

Note: Sierpinski conjectured a long time ago that every integer can be expressed as a sum of four cubes. This has been verified up to about 107 (Richard Lukes).

It has also been conjectured that every integer has the form x3 + y3 + 2 z3. For that problem, the first cases for which no expression is known are 148, 671, and 788. If you can find solutions to any of these, let me know.

Of course, this problem requires computer assistance. And the integers being cubed can be negative or zero.

In case you are wondering, the problem number is a sum of three positive cubes: 1080 = 93 + 73 + 23

Suggested by Larry Carter (Univ. of Calif., San Diego), with additional comments by Richard Guy (Univ. of Calgary).

© Copyright 2007 Stan Wagon. Reproduced with permission.

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18 September 2007