一类超椭圆曲线的整点个数

The Number of Integral Points on a Class of Hyperelliptic Curves

设ｎ是大于２的工整数，Ｄ是无平方因子正整数，分别是Ｋ的理想类群和类数．对于正整数ｍ，设ｇｋ（ｍ）是Ｉｘ中阶数等于ｍ的理想类的个数．本文证明了：超椭圆曲线ｆ（ｘ，ｙ）＝Ｄｘ２－４ｙｎ＋１＝０上整数点（ｘ，ｙ）的个数不超过ｍａｘ（８，２１６４Ｐ８１ｇｋ（Ｐ）），其中ｐ是ｎ的奇素因数．

Let n be a positive integer with n > 2. Let D be a positive integer with square free,and let Ik, hk denote the ideal class group and the class number of the imaginary quadratic field K=respectively.In this paper we prove that the number of integral points (x, y) on the hyperelliptic curve f(x, y) =-Dx2-4yn+1 is no more than max (8, 2164p81gx(p)) for any odd prime factor p of n, where gK(p) is the number of ideal classes in Ik with order p.