Consecutive and Parity-Consecutive Complete Lucas Sequences when the Period Equals the Modulus The Lucas sequence of the first kind (LSFK) is denoted (Un(p,q))n≥0, where Un is its nth term. The LSFK is defined recursively by Un = pUn−1−qUn−2 with initial terms U0 = 0 and U1 = 1, for integers p and q. The period length of (Un(p,q))n≥0 (mod m), denoted π(m), is the least positive integer r such that Un+r ≡ Un (mod m) for all n ≥ 0. Although the distribution of sequences has been studied, none have classified the conditions when each residue occurs exactly once. This motivates the following definitions. A sequence is complete when π(m) = m and the m repeating terms of (Un(p,q) (mod m))π(m)−1 n=0 are some permutation of the values 0,1,2,...,m − 1. Investigation of complete sequences reveals unique patterns. There exist complete consecutive sequences where Un ≡ n (mod m) for all 0 ≤ n ≤ m−1. Furthermore, parity-consecutive complete sequences exist, in which (Un(p,q))n≥0 (mod m) decomposes into the disjoint subsequences (U2n(p,q))n≥0 and (U2n+1(p,q))n≥0 modulo m containing all even and odd terms, respectively. Our research determines the values of p and q that yield the various forms of completeness under certain moduli m
