Prime-Modulus Periodicity of Multiplicatively Weighted Lucas Sums

Abstract. We study the modular periodicity of partial sums of Lucas sequences weighted by a multiplicative pseudorandom sequence. Specifically, for an odd prime $p$ and integers $P, Q$ with $p \nmid Q$, let $U_n(P,Q)$ be the Lucas sequence of the first kind and let the weights $a_n$ be generated by the multiplicative rule $a_n \equiv k^{n-1} \pmod t$, where $t \ge 2$ and $\gcd(k, t) = 1$. We consider the weighted partial sums $S_m = \sum_{n=1}^m (a_n \bmod p) U_n \bmod p$. First, we construct a finite-state dynamical system and prove that the sequence $\{S_m\}$ is purely periodic. Second, by analyzing the drift over a common period of the increment sequence, we show that the least period $\tau$ divides $p \cdot \operatorname{lcm}(\pi(p), \lambda)$, where $\pi(p)$ is the period of $U_n$ modulo $p$ and $\lambda = \operatorname{ord}_t(k)$. Moreover, if the drift vanishes, we obtain the sharper bound $\tau \mid \operatorname{lcm}(\pi(p), \lambda)$. Third, under the additional assumptions $\gcd(\pi(p), \lambda) = 1$, vanishing drift, $p \nmid \pi(p)\lambda$, and non-collapsing of the reduced weight period, we establish a spectral criterion: $\tau$ attains the maximal possible value $\pi(p)\lambda$ if and only if the greatest common divisor of the Fourier support of the increment sequence together with $\{\pi(p)\lambda\}$ equals $1$. The same framework extends immediately to Lucas sequences of the second kind. The results provide a rigorous prime-level foundation for the period analysis of weighted Lucas sums.