the Ramanujan machine

Nature of 4 Feb. 2021 offers a rather long (Nature-like) paper on creating Ramanujan-like expressions using an automated process. Associated with a cover in the first pages. The purpose of the AI is to generate conjectures of Ramanujan-like formulas linking famous constants like π or e and algebraic formulas like the novel polynomial continued fraction of 8/π²:

\frac{8}{{{\rm{\pi }}}^{2}}=1-\frac{2\times {1}^{4}-{1}^{3}}{7-\frac{2\times {2}^{4}-{2}^{3}}{19-\frac{2\times {3}^{4}-{3}^{3}}{37-\frac{2\times {4}^{4}-{4}^{3}}{\ldots }}}}

which currently remains unproven. The authors of the “machine” provide Python code that one can run to try uncover new conjectures, possibly named after the discoverer! The article is spending a large proportion of its contents to justify the appeal of generating such conjectures, with several unsuspected formulas later proven for real, but I remain unconvinced of the deeper appeal of the machine (as well as unhappy about the association of Ramanujan and machine, since S. Ramanujan had a mystical and unexplained relation to numbers, defeating Hardy’s logic,  “a mathematician of the highest quality, a man of altogether exceptional originality and power”). The difficulty is in separating worthwhile from anecdotal (true) conjectures, not to mention wrng conjectures. This is certainly of much deeper interest than separating chihuahua faces from blueberry muffins, but does it really “help to create mathematical knowledge”?

4 Responses to “the Ramanujan machine”

  1. […] was reminded of this topic yesterday when reading this comment by X, so I googled *Is sqrt(2) a normal number?* and came across the above-linked […]

  2. Some of the comments on this blogpost might be worth noting, especially the one concerning the algorithms used.

    • Thank you for the link, the post provides stronger arguments about the limitations of this paper.

    • A comment from Chitro Majumdar: Ramanujan got his formulas through thinking and because he had a good feeling. When Hardy first saw Ramanujan’s written notes he said something like this: some of the formulas we know, some of them we don’t and I don’t see how to prove them but even the things that are known, Ramanujan (re)discovered them by methods we don’t know or understand. He then invited Ramanujan to the UK. Ramanujan had theories in his mind that produced the outcomes. No experiments, no guessing. Without having a clue on what today is called algebraic geometry, he made fundamental contributions in this field.

      The project they talk about in Nature is different and has nothing to do with Ramanujan’s methods or thinking. They produce continued fractions and if there is a regularity it leads to a conjecture. This is experimental work which was not possible 10 years ago. Understanding these regularities is something that is hard. There are numerous (seemingly regular) expansions for e and pi and their combinations. Some of them we do not understand. It offers a lot of new questions and since there is no theory behind the regularity in continued fraction expansions, the questions might remain open for some time. Some of the regularities might break down if a million more steps are taken. Such things happen in number theory.

      In the same style: it is still unknown whether e or pi are “normal” numbers. Nevertheless the set of normal numbers has full measure, the complement has zero Lebesgue measure.

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