Divisibility and Modular Periodicity of a Prime-Discriminant Lucas Sequence Modulo Prime Powers

This study examines the divisibility and modular periodicity of a restricted Lucas sequence whose discriminant is a prime congruent to one modulo four. A congruence relation is derived modulo the associated prime and used to prove that the prime divides a sequence term if and only if it divides the corresponding index. This establishes the rank of apparition, and the period modulo the prime is also derived. The investigation then tests an extension of the period formula to higher prime powers using Python. All 80 eligible primes below 1000 were examined for exponents one through four, giving 320 different cases. Candidate periods were checked using modular matrix exponentiation, and their minimality was tested by examining divisors obtained from their prime factors. In every tested scenario, the period followed a pattern determined by the prime power and the multiplicative order of four modulo the prime. These computations support the proposed extension over the specified range, without representing a general proof for higher prime powers.