CAO Rui,LUO Ming.On the Diophantine Equation x3±1=1 379 y2[J].Journal of Chongqing Technology and Business University(Natural Science Edition）,2020,37(4):118-122

On the Diophantine Equation x3±1=1 379 y2

DOI：

 作者 单位 曹瑞， 罗明 重庆师范大学 数学科学学院,重庆 401331

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=1 379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=1 379y2仅有整数解(x,y)=(－1,0),不定方程x3－1=1 379y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=1379y2has only integer solutions (x,y)=(－1,0)，and that the Diophantine equation x3－1=1379y2 has only integer solutions (x,y)=(1,0).The algebraic method used can be extended to solve cubic Diophantine equations with large coefficients.