Machine learning applications in number theory Is there any research into or applications of machine learning in number theory?
I am also looking for (leading examples of) statistical/empirical analysis of number theory questions. Also wondering if genetic algorithms in particular have ever been used in these areas.

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*roughly related question on other site: Why can machine learning not recognize prime numbers?


*an area in number theory that seems to have had some statistical analysis, the Collatz conjecture.


*possibly somewhat related, automated theorem proving.
 A: Genetic algorithms were used to lower the prime gap to 4680 in the recent Zhang twin primes proof breakthrough and associated Polymath project. The bound has been lowered by other methods but it shows some potential for machine learning approaches in this or related areas. they can be used to devise/optimize effective "combs" or basically sieves for analyzing/screening smallest-possible prime gaps.
Together and Alone, Closing the Prime Gap  (Erica Klarreich, Quanta magazine, 19 November 2013):

The team eventually came up with the Polymath project’s record-holder — a 632-tooth comb whose width is 4,680 — using a genetic algorithm that “mates” admissible combs with each other to produce new, potentially better combs.

A: See the 2019 preprint Machine Learning meets Number Theory:
The Data Science of Birch-Swinnerton-Dyer by Alessandretti, Baronchelli & He. Here is the Abstract:

Empirical analysis is often the first step towards the birth of a conjecture.
This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing
the rational points on an elliptic curve, one of the most celebrated unsolved
problems in mathematics. Here we extend the original empirical approach, to
the analysis of the Cremona database of quantities relevant to BSD, inspecting
more than 2.5 million elliptic curves by means of the latest techniques in data
science, machine-learning and topological data analysis.


Key quantities such as rank, Weierstrass coefficients, period, conductor,
Tamagawa number, regulator and order of the Tate-Shafarevich group give rise
to a high-dimensional point-cloud whose statistical properties we investigate.
We reveal patterns and distributions in the rank versus Weierstrass coefficients,
as well as the Beta distribution of the BSD ratio of the quantities. Via gradient
boosted trees, machine learning is applied in finding inter-correlation amongst
the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and
more in general in pure Mathematics.

