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 \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases} \end{equation} where $A,B,C,D,n \in\ZZ_{+}$ and $n\geq3$, 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 equation $(1)$ in quadratic fields $\mathbb{Q}(\sqrt{t})$ for $n=3$.