有关LUCAS序列的几个充要条件

SOME NECESSARY AND SUFFICIENT CONDITIONS ON LUCAS SEQUENCES

Lucas序列Un(u)和Vn(u)定义为:U0=0,V0=2,U1=1,V1=u,Un=uUn-1-Un-2,Vn=uVn-1-Vn-2,n≥2.本文分别给出了同余式组 UN+r(u)≡0modNVN+r(u) 2modN,UN+r(u) 0modNVN+r(u)≡2modN和UN+r(u) 0modNVN+r(u) 2modN成立的几个充要条件,并对满足同余式组的u的个数进行估计,其中N=pq是两个奇素数之积,q=k(p+1)+r,｜r｜<p+12,k≥7,(u2-4p)=-1且gcd(u,N)=gcd(u2-4,N)=1.

The lucas sequences U_n(u) and V_n(u) were defined as: U_0=0, V_0=2, U_1=1, V_1=u, U_n=uU_(n-1)-U_(n-2),V_n=uV_(n-1)-V_(n-2), n≥2. In this paper, we present some necessary and sufficient conditions on the systems of congruences: U_(N+r)(u)≡0 mod NV_(N+r)(u)2 mod NB),U_(N+r)(u)0 mod NV_(N+r)(u)≡2 mod N and U_(N+r)(u)0 mod NV_(N+r)(u)2 mod Nand estimate the number of u's satisfying the above systems respectively, where N=pq is a product of two odd primes, q=k(p+1)+r,|r|<p+12,k≥7,(u^2-4p)=-1, and gcd(u,N)=gcd(u^2-4,N)=1.