Research On Special Numbers In Linear Recurrence Sequences

The related issues of special number in linear recurrence sequence has been one of the important contents of number theory research. Many scholars have done a lot of in-depth researches about this issue and got many fruitful results. In 1996, P.Ribenboim and W.L.McDaniel have studied the square in the Lucas sequence of the first type Un(P, Q),2Un(P, Q) and Lucas sequence of the second type Vn(P,Q),2Vn(P,Q). In 2004, A.Bremner and N.Tzanakis have investi-gated for the values of the parameters P and Q, when n= 9,12 and Un(P, Q) is a square. In 2007, A.Bremner and N.Tzanakis have researched further on the values of the parameters n, P and Q on the basis of previous research, when the Lucas sequence of the first type Un(P, Q) is a square. In 2013, Francesca Balestrieri has studied the values of the parameters n, P and Q, when Un(P, Q) is not a square. In this paper, we research on the special number in linear recur-rence sequence and the period of linear recurrence sequence by using elementary tools, the main content of this paper is as follows:1. Let P, Q be integers, Un(P,Q) is the Lucas sequence of the first type. This paper have studied the values of the parameters n, P and Q, when Un(P, Q) is not the k-th power.2. Let P, Q, R be integers, Un(P,Q,R) is a new definition of rational recurrence sequence. This paper have researched on the values of the parameters n, P, Q and R, when Un(P, Q,R) is not a square.3. The period of linear recurrence sequence module 4.4. In this paper, we have determined which items in linear recurrence se-quence is not square or not the k-th power by restricting the values of parameters in linear recurrence sequence.