【摘要】 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)。更多还原

【Abstract】 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~2are 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.更多还原