关于三次Diophantine方程x^3-1=2py^2

On the cubic diophantine equation x^3-1=2py^2

文中利用初等方法以及同余理论,讨论三次Diophantine方程x3-1=2py2当p为适合p≡1(mod6)的奇素数时的可解性。给出了该方程有解的充要条件和推论,并且仅有正整数解(x,y)=(2a2+1,aB(4a4+6a2+3))及(x,y)=(6a2+1,3aB(12a4+6a2+1))。

In this paper some elementary methods and congruence theory are used to discuss the solvability of the cubic Diophantine equation x3 - 1 = 2py2 when p is an odd prime with p = 1 （ mod6 ）. A necessary and sufficient condition for the equation to have solutions and its corollaries are given, and the only positive integer solutions of the equation are （x,y） = （2a2 ＋ 1,aB（4a4 ＋6a2 ＋3）） and （x,y） = （6a2 ＋ 1,3aB（12a4 ＋6a2 ＋ 1））.