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 Q, let Uₙ (P, Q) be the Lucas sequence of the first kind and let the weights aₙ be generated by the multiplicative rule aₙ k^n-1 t, where t 2 and (k, t) = 1. We consider the weighted partial sums Sₘ = ₍=₁ᵐ (aₙ p) Uₙ p. First, we construct a finite-state dynamical system and prove that the sequence \Sₘ\ is purely periodic. Second, by analyzing the drift over a common period of the increment sequence, we show that the least period divides p lcm ( (p), ), where (p) is the period of Uₙ modulo p and = ordₜ (k). Moreover, if the drift vanishes, we obtain the sharper bound lcm ( (p), ). Third, under the additional assumptions ( (p), ) = 1, vanishing drift, p (p), and non-collapsing of the reduced weight period, we establish a spectral criterion: attains the maximal possible value (p) if and only if the greatest common divisor of the Fourier support of the increment sequence together with \ (p) \ 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.