一类不定方程的非负整数解

The Non-negative Integer Solution of A Type of Indeterminate Equations

定义k～1型广义Fibonacci数列,研究其通项与性质;结合相关文献方程和解的特点,猜想并证明了X2+kXY-Y2+1=0(k∈N*)这一类不定方程有且只有k～1型广义Fibonacci数列形式的非负整数解.

In this paper we define k -1 type of generalized Fibonacci series and discuss its general term and properties. By combining with the features of equation and reconciliation in the relevant literature, we pose a conjecture and prove that a type of indeterminate equations X^2 ＋kXY- y^2 ＋ 1 =0（k ∈N＊） has non-nega- tive integer solutions only in the form of k - 1 type of generalized Fibonaeci series.