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 20161110 21:46:05 