关于不定方程x^(3)-1=114y^(2)

On the Diophantine Equation x^(3)-1=114y^(2)

不定方程是数论中不可或缺的一个分支,它有着悠久的历史与丰富的内容,其理论和方法在各学科和实际生活中都有广泛的应用。运用同余式、递归序列、平方剩余以及Pell方程的解的性质等初等方法对不定方程x^(2)-1=114y^(2)的整数解进行了讨论。首先利用因式分解将原不定方程分解为8种情形,其次运用转化、取模等技巧对8种情形分别分析,最终得出不定方程x^(2)-1=114y^(2)仅有整数解(x,y)=(1,0)。

Diophantine equation,an indispensable branch of number theory,has a long history and rich content,and its theory and method are widely used in various disciplines and practical life.In this paper,the integer solutions of indefinite equation x^(2)-1=114y^(2)are discussed by means of congruence,recursive sequence,quadratic residue and properties of the solution of the Pell equation.Firstly,the original indefinite equation is decomposed into 8 cases by factorization.Secondly,the eight cases are analyzed by transformation and mod ulo taking techniques.Finally,the only integer solution(x,y)=(1,0)is obtained for the indefinite equation x^(2)-1=114y^(2).