We study lower bounds for the least period of multiplicatively weighted Lucas partial sums modulo an odd prime. In the companion paper, a general upper-bound theory was established for weighted sums Sm=∑n=1m (an mod p) Un (P, Q) in Fp, Sm=∑n=1m (anmodp) Un (P, Q) in Fp, where Un (P, Q) Un (P, Q) is the Lucas sequence of the first kind and an≡kn−1 (modt) an≡kn−1 (modt). That theory naturally raises the question whether the least period ττ admits any nontrivial lower bound in terms of the Lucas period π (p) π (p) and the ambient weight period λ=ordt (k) λ=ordt (k). We prove that, in complete generality, the answer is negative. More precisely, we construct an infinite family of parameters for which τlcm (π (p), λ) lcm (π (p), λ) τ can be made arbitrarily small. Equivalently, for every N≥1N≥1 there exist admissible parameters such that τ<1Nlcm (π (p), λ). τ<N1lcm (π (p), λ). Thus no nontrivial universal lower bound depending only on π (p) π (p) and λλ can hold. The mechanism is explicit: we force the reduced weight period λpλp to collapse to 11 while keeping the ambient period λλ arbitrarily large, and then exploit vanishing drift on the shorter increment block. This shows that the upper-bound theory from the companion paper is genuinely one-sided: the ambient period λλ can be almost invisible from the viewpoint of the modular least period.
Jianming Wang (Sat,) studied this question.