Shor’s algorithm is one of the most well-known applications in quantum computing, capable of efficiently factoring large semiprime numbers and thus highlighting potential vulnerabilities in classical cryptosystems such as RSA. The algorithm relies on the rapid determination of the period r of the modular exponentiation function f(x)= a^x mod N modn. Since the success of the algorithm depends critically on obtaining the correct period, understanding how these periods are distributed across different semiprimes may offer deeper insight into both the theoretical behavior and the practical efficiency of Shor’s algorithm. In this study, we examine semiprimes formed from pairs of primes ranging from 3 to 103 and analyze the distribution of their corresponding maximum period values r. Our results indicate that the period values associated with different prime pairs follow an approximately linear pattern when plotted in relation to their underlying primes. Notably, prime combinations that lie along lines with smaller slopes tend to yield shorter periods. This observation suggests that semiprimes exhibiting such characteristics may correspond to cases in which Shor’s algorithm identifies the period more efficiently, potentially making those semiprimes more susceptible to quantum factorization. Beyond identifying these linear structures, our analysis also highlights that the relationship between the maximum period and the size of the semiprime is not random but reveals clear mathematical regularities. These patterns may serve as indicators for estimating the relative difficulty of factoring certain semiprimes under Shor’s algorithm. In the broader context of cryptographic security, such findings may contribute to the evaluation of RSA key strength and help guide the selection of more robust prime candidates in anticipation of future quantum-computing capabilities.