My latest column for Quanta Magazine is about the search for sums of cubes. While most integers are neither cubes nor the sum of two cubes, it is conjectured that most numbers can be written as the sum of three cubes. Finding those three cubes, however, can be a challenge.
For example, it was only this year we learned that the number 33 could be written as a sum of three cubes:
33 = 8,866,128,975,287,528³ + (−8,778,405,442,862,239)³ + (−2,736,111,468,807,040)³
What’s so hard about expressing numbers as a sum of three cubes?
It’s not hard to see that it combines the limited choices of the sum-of-squares problem with the infinite search space of the sum-of-integers problem. As with the squares, not every number is a cube. We can use numbers like 1 = 1³, 8 = 2³ and 125 = 5³, but we can never use 2, 3, 4, 10, 108 or most other numbers. But unlike squares, perfect cubes can be negative — for example, (-2)³ = -8, and (-4)³ = -64 — which means we can decrease our sum if we need to. This access to negative numbers gives us unlimited options for our sum, meaning that our search space, as in the sum-of-integers problem, is unbounded.
To learn more, read the full article, which is freely available here.
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