Verification of Prime Periodicity Structure Based on Quantum Phase Estimation and Classical Fourier Analysis and a Composite Generation Model

Shor's algorithm proved that quantum computers can efficiently factor integers by finding the period of a modular exponential function, with Quantum Phase Estimation (QPE) as its core. This paper proposes a novel “prime wave superposition” framework: each prime is regarded as the peak of a periodic cosine wave, and when the peaks of multiple waves coincide, the corresponding integer 30k+1 is bound to contain these primes as factors. This constructive property is used to generate composites systematically. We use the pyQPanda quantum virtual machine to build the unitary operator +1 mod p and autonomously discover the eigen-periods for moduli p = 7, 11, 31, 61 via QPE, providing physics‑based verification of each prime period without relying on a priori formulas. Furthermore, the Chinese Remainder Theorem (CRT) is employed to precisely compute the coincidence positions k of multiple wave peaks, yielding composites X = 30k+1 whose classical factorization confirms that they indeed contain the corresponding prime factors. Building upon this, we introduce classical Discrete Fourier Transform (DFT) into the framework. By performing spectral analysis on a binary “prime divisibility” signal, we independently extract all four prime periods and observe harmonic coincidences in the frequency domain. In parallel, a classical Shor‑FFT method is applied to factor the CRT‑generated composites, with a recursive strategy achieving complete prime factor lists. Additionally, we implement the full quantum Shor algorithm on pyQPanda, successfully factoring 91 = 7 * 13 and 1891 = 31 * 61 on a genuine quantum circuit, thereby providing direct quantum evidence. Notably, our custom Shor implementation succeeded on 1891, whereas the general‑purpose Shor API shipped with pyQPanda is unable to handle this number due to conservative resource estimation. This contrast further underscores the practicality of our approach.The integration of quantum period discovery, classical Fourier analysis, number‑theoretic CRT construction, and quantum Shor experiments forms a multi‑layered verification system, offering a comprehensive experimental platform for teaching Shor's algorithm, demonstrating quantum advantage in miniature, and exploring prime‑sieve models.