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Machine learning of the prime distribution.

Alexander Kolpakov1, A Alistair Rocke2

  • 1University of Neuchâtel, Neuchâtel, Switzerland.

Plos One
|September 27, 2024
PubMed
Summary
This summary is machine-generated.

This study applies maximum entropy methods to probabilistic number theory, yielding theorems like the Hardy-Ramanujan Theorem. It also explains prime learnability and suggests machine learning may miss the Erdős-Kac law.

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Area of Science:

  • Number Theory
  • Probability Theory
  • Machine Learning Theory

Background:

  • The Hardy-Ramanujan Theorem describes the distribution of additive number theoretic functions.
  • Understanding the learnability of prime numbers is crucial for theoretical computer science.
  • The Erdős-Kac law concerns the distribution of prime factors of integers.

Purpose of the Study:

  • To derive new theorems in probabilistic number theory using maximum entropy methods.
  • To provide a theoretical explanation for observed phenomena in prime number learnability.
  • To assess the likelihood of discovering the Erdős-Kac law with contemporary machine learning algorithms.

Main Methods:

  • Application of maximum entropy methods for theorem derivation.
  • Theoretical analysis of prime number learnability.
  • Comparative analysis of theoretical findings with machine learning capabilities.

Main Results:

  • Several theorems in probabilistic number theory were derived, including a novel version of the Hardy-Ramanujan Theorem.
  • A theoretical framework was established to explain experimental observations on prime learnability.
  • It was determined that the Erdős-Kac law is unlikely to be discovered by current machine learning techniques.

Conclusions:

  • Maximum entropy methods are effective for advancing probabilistic number theory.
  • The learnability of primes has a theoretical basis that current AI may not readily uncover.
  • The limitations of machine learning in discovering fundamental mathematical laws like the Erdős-Kac law are highlighted.