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Sums of three cubes

Is there a number (that is not 4 or 5 modulo 9) that cannot be expressed as a sum of three cubes?

Put another way :

In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for n to equal such a sum is that n cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and โˆ’1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient."

Source : Wikipedia

The solutions for most numbers up to 1000 have been found, but 114, 390, 627, 633, 732, 921 and 975 remain. After 1000, there is, to date, no proof one way or the other.


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