Success probability in Shor's Algorithm

This paper aims to determine the exact success probability at each step of Shor's algorithm. Although the literature usually provides a lower bound on this probability, we present an improved bound. The derived formulas enable the identification of all failure cases in Shor's algorithm, which correspond to a success probability of zero. A simulation routine is provided to evaluate the theoretical success probability for a given integer when its prime factorization is known with potential applications in quantum resource estimation and algorithm benchmarking.