关于指数丢番图方程$x^2+(3a^2-1)^m=(4a^2-1)^n$

On the Exponential Diophantine Equation $x^2+(3a^2-1)^m=(4a^2-1)^n$

中文关键词:  指数丢番图方程  Lucas序列  本原素因子  Kronecker符号.

英文关键词:exponential Diophantine equations  Lucas sequences  primitive divisors  Kronecker symbol.

中文摘要:

      应用Bilu, Hanrot 和 Voutier关于本原素因子的深刻结果以及二次丢番图方程解的表示的一些精细结果,完全解决了指数型丢番图方程$x^2+(3a^2-1)^m=(4a^2-1)^n$当$3a^2-1$是奇素数或奇素数幂时的求解问题.

英文摘要:

      We apply a new, deep theorem of Bilu, Hanrot \& Voutier and some fine results on the representation of the solutions of quadratic Diophantine equations to solve completely the exponential Diophantine equation $x^2+(3a^2-1)^m=(4a^2-1)^n$ when $3a^2-1$ is a prime or a prime power.