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Old 2020-09-20, 20:09   #1
(loop (#_fork))
fivemack's Avatar
Feb 2006
Cambridge, England

143578 Posts
Default largest n such that n^2+1 has prime factors within a set

To find Machin-like 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^2-1 gives you combinations of hyperbolic-arc-cotangents which sum to the logarithms of small primes, which combined with efficient series-summing tools give quite good expressions for said logarithms - I've used y-cruncher 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, y-cruncher has quite a large per-term overhead, so getting faster convergence for the individual arctangents doesn't win if you have to sum more terms; so

? 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]~
isn't actually useful)
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