Consistency Verification of Composite Number Factorization Using the Modular Sieve Method on Classical Computers and Quantum Simulators

Based on the modular sieve method (8-orbit structure, logarithmic decay law) independently developed by the author, this paper implements composite number factorization on both classical computers and quantum simulators, and demon- strates that the results from both platforms are completely consistent. Classical verification was performed on a 2013 Mac Pro, where 9-digit composite numbers were factorized in an average of 5 seconds, and the Hardy–Littlewood conjecture was numerically verified up to 100 billion in 878.8 seconds. Quantum verifi- cation used a 20‐qubit circuit, successfully factorizing the composite number 1891 on pyqpanda in 18 seconds, with validation also performed on Qiskit. The two inde- pendent verifications corroborate each other, providing a solid empirical foundation for the correctness and lightweight nature of the modular sieve method.