关于丢番图方程x^2-Dy^4=1的一些注记

SOME REMARKS ON THE DIOPHANTINE EQUATION x 2-Dy 4=1

对于丢番图方程ｘ２－Ｄｙ４＝１（Ｄ＞０且不是平方数），本文得到了如下结论：①当Ｄ＝ｐ是素数，且ｐ３，１５（ｍｏｄ１６）时，方程除开Ｄ＝５有解（ｘ，ｙ）＝（９，２）外，无其他的正整数解；②当Ｄ＝ｐ是素数时，方程除开Ｄ＝３有两组解（ｘ，ｙ）＝（２，１），（７，２）外，最多有一组正整数解；③当Ｄ＝４ｐ且奇素数ｐ１ｍｏｄ１６）时，方程除开Ｄ＝２０有解（ｘ，ｙ）＝（１６１，６）外，无其他的正整数解；④当Ｄ＝１６ｐ，ｐ是奇素数时，方程除开Ｄ＝４８，８０均有解ｙ＝１外。

In this paper, the following conclusions are proved:① If p is prime and p3, 15(mod16), then the Diphantine equations x 2-py 4=1 has no positive integer solutions (x,y) except 9 2-5.2 4=1.②If p is prime, then the Diophantine equation x 2-py 4=1 has at most one positire integer solutions (x,y),except 2 2-3·1 4=1 and 7 2-3·2 4=1.③If p is prime and p1 (mod16), then the Diophantion equations x 2-4py 4=1 has no positive integer solutions (x,y), except 161 2-4·5·6 4=1.④If p is prime, then the Diophantine equations x 2-16py 4=1 has no positive integer solutions (x,y), except 7 2-16·3·1 4=1 and 9 2-16·5·1 4=1.