A new generalization of Fermat's Last Theorem

Abstract

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(1){A+B=CABC=Dn where A,B,C,D,n∈Z+ and n≥3, which may be regarded as a generalization of Fermat's equation xn+yn=zn. 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)=pk 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(t) for n=3.