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2020-09-21, 03:23   #4
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

100100110101112 Posts
Quote:
 Originally Posted by fivemack To find Machin-like formulae for pi, I want to find sets of N where N^2+1 has only small prime factors. Tangentially, ...
Ah, it brings so many childhood memories. The year was 1980 and I thought it was interesting to compute many digits of π. Because I was only in 8th grade, my first implementation was based on $$6 \ atan {1 \over \sqrt 3}$$. believe it or not; and it worked of course with a proper implementation of a long $$\sqrt 3$$, but was relatively slow but I probably got 10,000 digits or so. Only then I learned form an encyclopedia about Machin's $$4 ( 4 \ atan {1 \over 5} - atan {1 \over 239})$$ (imagine the world without internet, heh?), and I remember how I could not believe my eyes and then checked that it was actually true (use $$tan$$ of both sides, and $$tan$$ of a sum of angles repeatedly), and spent some time searching for better variants but my foundation was too weak to make any progress except for brute force. I did get 100,000 decimal digits on BESM-6's using Algol code that my father ran at work at the Nuclear Center as his own.

Ah memories, memories...

I remember that I submitted that computation to a school informatics (which was just beginning in the USSR) conference -- and went on to present it in my first talk in my life in 10th grade - and the other viral problem I learned from a talk next to mine was the 'couch problem' where another kid was just cutting pieces from a digital rectangle, and he didn't get much far but he knew (and put it in his talk) at that time the best known answer of $$S = {\pi \over 2} + {2 \over \pi}$$ .and I remember being able to get to that answer analytically because at that time I already knew derivatives and some trigonometry.