关于Lucas序列中渐近平方数的研究

Research on near squares in a Lucas sequence

运用初等数论方法和四次Diophantine方程的结论讨论了不定方程vk=s2±1的可解性,并给出该方程的所有正整数解(k,s),其中vk=αn-αn/α-α,k是任意非负整数,α=a+a2-1(1/2),α=a-(a2-1)(1/2),a>1为固定的正整数.

By using the elementary number theory methods and some results on the number of quartic Diophantine equations,the solvability of the indeterminate equation vk=s^2±1 is dis-cussed,and its all positive integer solution （k,s）are completely determined,where vk=^{α^n-α^n}/α-α,k is any nonnegative integer,α=a＋√a^2-1,α=a-√（a^2-1）,a〉1,a is a fixed positive integer with a＆gt;1 .