20200920, 20:09  #1 
(loop (#_fork))
Feb 2006
Cambridge, England
2·11·17^{2} Posts 
largest n such that n^2+1 has prime factors within a set
To find Machinlike formulae for pi, I want to find sets of N where N^2+1 has only small prime factors. Tangentially, it would be nice to have a proof that, for example, n=485298 is the largest number where n^2+1 has no prime factor greater than 53.
(asking the same question about n^21 gives you combinations of hyperbolicarccotangents which sum to the logarithms of small primes, which combined with efficient seriessumming tools give quite good expressions for said logarithms  I've used ycruncher to get ten billion digits of log(17) without difficulty) And I've not the faintest clue where to start for this sort of question. (annoyingly, ycruncher has quite a large perterm overhead, so getting faster convergence for the individual arctangents doesn't win if you have to sum more terms; so Code:
? lindep([Pi/4,atan(1/485298),atan(1/85353),atan(1/44179),atan(1/34208),atan(1/6118),atan(1/2943),atan(1/1772),atan(1/931)]) %64 = [1, 183, 215, 71, 295, 68, 163, 525, 398]~ ? lindep([Pi/4,atan(1/485298),atan(1/330182),atan(1/114669),atan(1/85353),atan(1/44179),atan(1/34208),atan(1/12943),atan(1/9466),atan(1/5257)]) %80 = [1, 808, 1389, 1484, 2097, 2021, 1850, 1950, 398, 2805]~ 
20200920, 20:17  #2  
"Robert Gerbicz"
Oct 2005
Hungary
1397_{10} Posts 
Quote:
This problem is solvable by https://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem using a Pell type equation, where on the right side there is a 1 not 1. 

20200920, 20:33  #3 
"Robert Gerbicz"
Oct 2005
Hungary
575_{16} Posts 
See also for lots of big arctan formulas: http://www.machination.eclipse.co.uk/FSChecking.html
(last update was at 2013) 
20200921, 03:23  #4  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2·3^{3}·13^{2} Posts 
Quote:
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. 

20200921, 09:52  #5  
(loop (#_fork))
Feb 2006
Cambridge, England
2·11·17^{2} Posts 
Quote:
Some of the example Machin formulae on Wikipedia clearly use a onelargeprime method which I can't find a very good description for  I've added some paragraphs to make the examples on https://en.wikipedia.org/wiki/Machinlike_formula a bit less unmotivated. 

20200921, 17:39  #6 
Romulan Interpreter
Jun 2011
Thailand
2^{5}×3×7×13 Posts 
About that age (a bit younger actually), one colleague of mine and me, after we learned from the teacher that 22/7 is a "good" approximation of \(\pi\) (known to the old Greeks too, of course), we started a "quest" to find better approximations, by increasing the denominator little by little and looking for a "suitable" numerator. No computers, all on paper, and with "pocket" calculators (which also, were kind of huge and expensive toys at the time, we call them pocket today, but in those times they were not called so, and you needed a backpack to carry them, but well... both our fathers had some accountingrelated jobs...). We did this "research" for few weeks, actually, and we found few "better" approximations, of which we were very proud, and almost ready to show them to the teacher, when we realized suddenly that you can get more accurate approximations, in fact as accurate as you want, just by writing first n digits of \(\pi\) over the corresponding power of 10, and reducing the fraction.
This is not a joke, haha, we were soooooooo disappointed! Last fiddled with by LaurV on 20200921 at 17:48 
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