Quote:
Originally Posted by fivemack
To find Machinlike 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 8
^{th} 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 BESM6'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 10
^{th} 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.