Algorithms of the Möbius function by random forests and neural networks

Abstract

The Möbius function \(\mu (n)\) is known for containing limited information on the prime factorization of n. Its known algorithms, however, are all based on factorization and hence are exponentially slow on \(\log n\). Consequently, a faster algorithm of \(\mu (n)\) could potentially lead to a fast algorithm of prime factorization which in turn would throw doubt upon the security of most public-key cryptosystems. This research introduces novel approaches to compute \(\mu (n)\) using random forests and neural networks, harnessing the additive properties of \(\mu (n)\). The machine learning models are trained on a substantial dataset with 317,284 observations (80%), comprising five feature variables, including values of n within the range of \(4\times 10^9\). We implement the Random Forest with Random Inputs (RFRI) and Feedforward Neural Network (FNN) architectures. The RFRI model achieves a predictive accuracy of 0.9493, a recall of 0.5865, and a precision of 0.6626. On the other hand, the FNN model attains a predictive accuracy of 0.7871, a recall of 0.9477, and a precision of 0.2784. These results strongly support the effectiveness and validity of the proposed algorithms.