Theorem. In a hypothetical equality A n + B n - C n = 0 in a number system with a prime base n > 2, mutually prime natural numbers A, B, C (mod n k) starting from k=1 equal to the numbers A 1 n^ (k-1), B 1 n^ (k-1), C 1 n^ (k-1) (mod n k), where k can be arbitrarily large.
Victor Sorokine (Thu,) studied this question.