A new generalization of Fermat's Last Theorem

作者：Cai, Tianxin^*; Chen, Deyi; Zhang, Yong

来源：Journal of Number Theory, 2015, 149: 33-45.

DOI：10.1016/j.jnt.2014.09.014

• Fermat's Last Theorem
• Additive and multiplicative functions
• Quadratic fields
• Elliptic curves

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.

• 出版日期2015-4
• 单位浙江大学