丢番图方程x^3-1=pQy^2的整数解的研究

ON THE DIOPHANTINE EQUATION x^3-1=pQy^2

主要利用同余式、二次剩余、Legendre符号的性质等初等方法证明了p≡7(mod24)为奇素数,Q=13,109,181,229,277,421,(p/Q)=-1时,丢番图方程x3-1=p Qy2仅有整数解(x,y)=(1,0).

By using congruence, quadratic residue and the natures of Legendre symbol to prove that the Diophantine x 3 equation x -1= pQy2 only has integer solution （x,y） = （1,0） when Q = 13,109,181,229,277,421 , p be odd prime withp ≡7（mod24）and （P/Q） = - 1 .