| In this paper, on the base of the period k(p) of the generalized Fibonacci sequence modulo a prime, we study the more general linear recursive sequence, more precisely,In Chapter1, we divide it into two main parts. In part1, we introduce necessary preliminaries which are used in this paper, in part2. we give the research summary and conclusions which are acquired in this paper.In Chapter2, In order to characterize properties of the period k(p) of the linear recursive sequence modulo a prime, we study the period T(p) of the corresponding characteristic matrix U modulo a prime, and we get the following main conclusions (set△=A2+4B):In the case where En+1=AEn+BEn-1+C0+C1n+C2n2+…+Ctnt we haveTheorem2.4.1Let p be a prime(p is the prime which is not less than t+2),(1) if (△/p)=0, then T(p)=p·ord(A/2);In particular, if (△/p)=0and A≡2(mod p), then T(p)=p;(2) if (△/p)=1, when A+B≡1(mod p), T(p)=p·ord(-B);when A+B(?)1(mod p), p·ord(-B)|T(p)|p(p-1);(3) if (△/p)=-1, then T(p)|2p(p+1)ord(B2).Theorem2.5.1Let p be an odd prime,The first category, when the characteristic polynomial g(x) has no multiple root, we have(1)when all the three roots belong to Fp, then T(p)|(p-1);(2)when only one of the three roots belongs to Fp, then T(p)|(p2-1);(3)when none of the three roots belong to Fp, then T(p)|(p3-1).The second category, when the characteristic polynomial g(x) has multiple root, we have(1)when the polynomial has triple root α, then the triple root must belong to Fp, and T(p)=p·ord(a);(2)when the polynomial has double root, the the double root must belong to Fp, and T(p)|p(p-1);Inference2.5.2Let p be an odd prime, In the case where En+2=AEn+1+BEn+CEn-1we have(l)if B=1, then T(p)=2p;(2)if B≠1, when (B/p)=1,T(p)=2ord((?));(B/p)=-1,ord(-B)|T(p)|(p2-1);(B/p)=0,T(p)=1... |
