关于不定方程x^3±1=1379y^2

On the Diophantine Equation x^3±1=1379 y^2

关于x^3±1=Dy^2(D>0)型不定方程的解法还没有一般性的结论;研究D=1379时不定方程x^3±1=Dy^2的可解性问题,利用同余理论、递归序列、平方剩余以及Pell方程解的性质证明了不定方程x^3+1=1379y^2仅有整数解(x,y)=(-1,0),不定方程x^3-1=1379y^2仅有整数解(x,y)=(1,0);所使用的代数方法可以推广到求解大系数的三次不定方程中去.

There is no general conclusion about the solution of x^3±1=Dy^2(D>0)type Diophantine equation.The solvability of x^3±1=Dy^2 for Diophantine equation when D=1379 is studied.By using congruence,recursive sequence,quadratic remainder and some properties of solutions of Pell equations,it is proved that the Diophantine equation x^3+1=1379 y^2 has only integer solutions(x,y)=(-1,0),and that the Diophantine equation x^3-1=1379 y^2 has only integer solutions(x,y)=(1,0).The algebraic method used can be extended to solve cubic Diophantine equations with large coefficients.