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

[israel at math.ubc.ca]

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:

...
>An (incomplete and very sketchy) explanation for the origin and
>existence of such relations is as follows:
>
>The formula:
>
>arctan(x)=(log(1+ix)-log(1-ix))/(2i)
>
>shows that we have to consider prime factors
>of 1+n^2 for natural integers n
>(such factors are of course exactly 2 and
>all primes congruent to 1 modulo 4).
>
>Consider variables representing logarithms 
>of the underlying complex primes
>in Gaussian integers. (one needs two variables 
>$x_p,y_p$ for every real prime).
>
>Linear combinations of arctan(n)^2
>can then be interpreted as 
>quadratic forms (over the gaussian integers
>without error) in these variables 
>which simplify to zero.
>
>The question of an infinity of such relations
>can probably be adressed arithmetically
>(I guess the answer is yes).
>
>There are less relations than among 
>linear combinations of arctangents
>(for linear relations among arctan(n)
>you get an additional relation every
>time all prime-factors of 1+n^2
>occured already previously and the coefficients
>can be deduced from prime-factorizations of 1+j^2)
>since we have to deal with complex coefficients
>(and there is perhaps also
>pi coming in, I did not think it through 
>completely).
>
>From a computational point of view,
>I generated all integers $n$ up to some 
>bound with 1+n^2 involving only small 
>primes.
>
>One searches then for a relation
>using for example the PSLQ algorithm 
>under maple (I did it using 1500
>Digits of precision).
>
>Pushing the study a little bit further,
>It should be possible to write down 
>everything theoretically using 
>factorisation in Gaussian integers.
>
>Best wishes,  Roland Bacher