UMA CARACTERIZAÇÃO DO MÍNIMO MÚLTIPLO COMUM POR PERÍODOS DE 2-SEQUÊNCIAS

Abstract

We characterize the least common multiple by means of the least period of a 2-sequence. For an integer m ≥ 4, we consider the periodic triangular sequence

      1, 2, . . . ,m − 1, m,m − 1, . . . , 3, 2, 1, 2, . . .

with period P = 2(m− 1) and the subsequence obtained by sampling every p-th term. We prove the exact criterion that a positive integer q is a period of the sampled sequence if and only if P | pq. If ρ denotes its least period, the divisibility characterization of the least common multiple yields pρ = lcm(p, P). Next, for arbitrary x, y ∈ N, we construct a 2-sequence with least period ρ2(x, y) such that lcm(x, y) = xρ2(x, y). This representation allows the standard properties of the least common multiple—symmetry, idempotence, homogeneity, monotonicity, associativity, and the prime-valuation description— to be reformulated in terms of these periods.

Keywords: 2-sequences; least common multiple; periodic sequences; divisibility.