UWEC CERCA 2026

The Lucas sequence of the first kind (LSFK), denoted Un(p, q)n≥0, is defined recursively by its nthterm, Un = pUn−1 − qUn−2 with the initial terms U0 = 0 and U1 = 1, for integers p and q. For anodd prime p, the sequence Un(p, q)n≥0 (mod p) is known to be uniformly distributed when pdivides the discriminant. A uniformly distributed LSFK containing each residue exactly once definesComplete sequences (CS). The conditions that result in CS have not previously been classified. Thestudy of Complete sequences leads naturally to the notions of Complete Consecutive sequences(CCS) and Parity Consecutive Complete sequences (PCCS). When Un(p, q)n≥0 is CompleteConsecutive for some modulo m, the moduli producing Complete sequences for the same p and qvalues depend entirely on m. Furthermore, the subsequences U−2n(p, q)n=0 and U2n+1(p, q)n=0 of aComplete Consecutive Un(p, q)n≥0 (mod m) directly relate to the disjoint subsequences of a samemodulo Parity Consecutive Complete sequence. These relations show the underlying conditionsthat produce the various forms of complete sequences, providing a complete classification of theiroccurrences. Our research has classified the values of p and q that yield CS, CCS, and PCCS undercertain moduli m.