Machine Intelligence, Ramanujan & Algebraic Geometry

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 (as I planned to write to the author) 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.

Years ago (50+), on the first of April Scientific American published a series of math’s curiosities. 

One was a counterexample for the 4-colour problem (the obvious way to start colouring led to an impossible chart, but nevertheless it was possible to colour it). Another one was a combination of e and pi that was equal to a natural number. Indeed it was for up to 15 decimals (at that time the border of what one could calculate), then there was not a zero. A math theorem said that such combinations approximate natural numbers in whatever precision you want. The example given was within reach of humans writing numbers, the theorem says there are numbers of this kind but maybe they have zillions of digits. There were other examples coming from physics. Of course April 1st !!!