关于不定方程x^4-Dy^2=1的一个注记

ON THE DIOPHANTINE EQUATION x 4-D·y 2 =1

设整数Ｄ＞０且不是平方数，本文证明了不定方程ｘ４－Ｄｙ２＝１除开Ｄ＝１７８５，４·１７８５，１６·１７８５时，分别有二组正整数解（ｘ，ｙ）＝（１３，４），（２３９，１３５２）；（ｘ，ｙ）＝（１３，２），（２３９，６７６）；（ｘ，ｙ）＝（１３，１），（２３９，３３８）外，最多只有一组正整数解（ｘ１，ｙ１），且满足ｘ２１＝ｘ０或２ｘ２０－１。

Ljunggren proved that the equationx 4-Dy 2=1(1)Where D >0 and is not a perfect square,has at most two solutions in positive integers. Results of various degrees have been found by many authors on equation(1). In this note,by utilizing Ko Terjanian Rotkiewicz method in Diophantine equations and a result in Ljunggren′s paper (1),we prove the following theore. Therome The equation (1) has at most one solution in positive integers except D =1785,4·1785,16·1785 in which the equation (1) has two positive integral solutions ( x,y) =(13,4),(239,1352),(x,y)=(13,2),(239,676),(x,y)=(13,1),(239,338) respectively.If the equation (1) has one solution ( x 1y 1) in positive integers,then x 2 1=x 0 or x 2 1=2x 2 0-1, where x 0+x 0)D is a fundamental solution of Pell equation x 2-Dy 2=1 .