Hello, as far as I know, the formula (13) on this paper shows a series for Catalan constant : https://hal.inria.fr/hal-00990465/document which is quite fast. It is used in the current y-Cruncher program and was used to compute hundreds of billion digits of it. 600 billion digits. here are the details : http://www.numberworld.org/digits/Catalan/ The rate of convergence of that series is big. Best regards, Simon Plouffe Le 2021-02-11 à 02:35, Ray Tayek a écrit :
https://science.slashdot.org/story/21/02/10/2056233/machines-are-inventing-n...
an anonymous reader quotes a report from Motherboard: /[A] group of researchers from the Technion in Israel and Google in Tel Aviv presented an automated conjecturing system that they call the Ramanujan Machine, named after the mathematician Srinivasa Ramanujan, who developed thousands of innovative formulas in number theory with almost no formal training. The software system has already conjectured several original and important formulas for universal constants that show up in mathematics <https://www.vice.com/en/article/xgzkek/machines-are-inventing-new-math-weve-never-seen>. The work was published last week in Nature <https://www.nature.com/articles/s41586-021-03229-4?utm_medium=affiliate&utm_source=commission_junction&utm_campaign=3_nsn6445_deeplink_PID100094349&utm_content=deeplink>.
One of the formulas created by the Machine can be used to compute the value of a universal constant called Catalan's number more efficiently than any previous human-discovered formulas. But the Ramanujan Machine is imagined not to take over mathematics, so much as provide a sort of feeding line for existing mathematicians. As the researchers explain in the paper, the entire discipline of mathematics can be broken down into two processes, crudely speaking: conjecturing things and proving things. Given more conjectures, there is more grist for the mill of the mathematical mind, more for mathematicians to prove and explain. That's not to say their system is unambitious. As the researchers put it, the Ramanujan Machine is "trying to replace the mathematical intuition of great mathematicians and providing leads to further mathematical research." In particular, the researchers' system produces conjectures for the value of universal constants (like pi), written in terms of elegant formulas called continued fractions. Continued fractions are essentially fractions, but more dizzying. The denominator in a continued fraction includes a sum of two terms, the second of which is itself a fraction, whose denominator itself contains a fraction, and so on, out to infinity.
The Ramanujan Machine is built off of two primary algorithms. These find continued fraction expressions that, with a high degree of confidence, seem to equal universal constants. That confidence is important, as otherwise, the conjectures would be easily discarded and provide little value. Each conjecture takes the form of an equation. The idea is that the quantity on the left side of the equals sign, a formula involving a universal constant, should be equal to the quantity on the right, a continued fraction. To get to these conjectures, the algorithm picks arbitrary universal constants for the left side and arbitrary continued fractions for the right, and then computes each side separately to a certain precision. If the two sides appear to align, the quantities are calculated to higher precision to make sure their alignment is not a coincidence of imprecision. Critically, formulas already exist to compute the value of universal constants like pi to an arbitrary precision, so that the only obstacle to verifying the sides match is computing time./