Primality test with pair of Lucas sequences

Abstract

Lucas sequences and their applications play vital role in the study of primality tests in number theory. There are several known tests for primality of positive integer $N$ using Lucas sequences which are based on factorization of $(N \pm 1)$ [2] [13]. In this paper we give a primality test for odd positive integer $N>1$ by using the set $L(\Delta, N)$ where $L(\Delta, N)$ is a set of $S(N)$ distinct pair of Lucas sequences $(V_n (a,1),U_n (a,1))$, where $S(N)$ for $N=p_1^{e_1}.p_2^{e_2}\ldots p_s^{e_s}$ is given as $S(N)=LCM[\{p_{i}^{e_{i}-1}(p_i-(\frac{\Delta}{p_i}))\}_{i=1}^s]$ and $\Delta=a^2-4$ for some fixed integer $a$.