[seqfan] Re: A sequence inspired by Størmer numbers (A005528).

[Joerg Arndt]

* israel at math.ubc.ca <israel at math.ubc.ca> [Oct 31. 2015 08:35]:
> Basically everything seems to come from the basic relations: if 1+ix =
> (a+ib)(c+id) then arctan(x) = arctan(b/a) + arctan(d/c) + n pi so that
> expressions in arctans of integers can be written in terms of arctans of
> rational numbers corresponding to Gaussian primes. These should essentially
> generate all the linear relations between arctans of rationals with rational
> coefficients. I don't know if all quadratic relations arise from these: I
> suspect it is not known, but might follow from Schanuel's conjecture (and
> for the purposes of OEIS, I think we can assume Schanuel's conjecture).
> 
> The use of PSLQ with a fixed precision may be rather dangerous: it's not
> clear if we are getting an exact relation or just a very good approximate
> one, or missing an exact relation with enormous coefficients.
> 
> Cheers,
> Robert
> 
> On Oct 30 2015, Roland Bacher wrote:
> [...]

Just a remark: the problem with the non-squared arctans
can be attacked via linear algebra, see Section 32.5
"Arctangent relations for Pi" in  http://jjj.de/fxt/#fxtbook

Best regards,   jj