N is a positive integer which is expressible as the sum of cubes of two positive integers.
Given that N is not divisible by 9, find the possible remainders when N is divided by 63.

(In reply to computer solution by Charlie)

I agree.

The cubes mod 63 have a period of 21, starting {1,8, 27,1,..}. We can sum all the pairs of remainders to produce a table, eliminating those divisible by 9, including 0.

This produces a symmetrical result; the required remainders are 63+/-1 {1,62}, 63+/-2 {2,61} etc. giving 63+/- {1,2,7,8,16,19,20,26,28,29} as the exhaustive list of solutions.

Posted by broll on 2016-11-10 21:46:05