UWEC CERCA 2026

The Lucas sequence of the first kind (LSFK) is denoted (Un(p, q))n≥0, where Un is its nth term.It is defined recursively by Un = pUn−1 − qUn−2 with initial terms U0 = 0 and U1 = 1, forintegers p and q. A sequence is uniformly distributed modulo m when each residue appears thesame number of times over the full period π(m), the number of repeating terms in any sequencemodulo m. However, the sequences modulo m in which each residue occurs exactly once withinthe full period are not classified. The following definitions for LSFK emerge from this observation.Complete sequences (CS) are defined by the two conditions: π(m) = m and the m repeating termsof (Un(p, q) (mod m))π(m)−1n=0 are some permutation of the values 0, 1, 2, . . . , m − 1. Completeconsecutive sequences (CCS) satisfy the congruence Un ≡ n (mod m) for all 0 ≤ n ≤ m − 1.Parity-consecutive complete sequences (PCCS) occur when (Un(p, q))n≥0 (mod m) decomposesinto the disjoint union of the two subsequences (U2n(p, q))n≥0 and (U2n+1(p, q))n≥0 modulo mcontaining all even and odd terms, respectively. The semi-quasi Lucas sequences (SQL) are notnecessarily complete, but they satisfy the relation, Un = Un−1 + Un−2 and Un + Un+1 = Un+2. Ourresearch determines the values of p and q that yield CS, CCS, PCCS, and SQL sequences undercertain moduli m.