A new generalization of Fermat's Last Theorem

In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation [GRAPHIC] where A, B, C, D, n is an element of Z(+) and n >= 3, which may be regarded as a generalization of Fermat's equation x(n) + y(n) = z(n). When gcd(A, B, C) = 1, (1) is equivalent to Fermat's equation, which means it has no positive integer solutions. We discuss several cases for gcd(A, B, C) = p(k) where p is an odd prime. In particular, for k = 1 we prove that (1) has no nonzero integer solutions when n = 3 and we conjecture that it is also true for any prime n > 3. Finally, we consider Eq. (1) in quadratic fields Q(root t) for n =3. (C) 2014 Elsevier Inc. All rights reserved.