[seqfan] Arc-Tangent Irreducible Rationals

[John Conway]

On Fri, 6 Jun 2003 pauldhanna at juno.com wrote:

> 
> Thank you for your kind reply - very informative and interesting.
> 
> Do "geodetic angles" address the hyperbolic version of Stormer numbers ("hyperbolic Stormer numbers", say),
 such that the arctanh of these numbers form a basis for the space of arctanh of rationals > 1? 

   Not as published, but I worked out the hyperbolic version of the more 
general theory I mentioned.  Namely, take the distances, which it's best 
to think of as being in hyperbolic space, whose squared hyperbolic 
functions are rational.  Then the theory tells you all the rational linear
relations between them.  

       Let me quickly comment, first on the angle theory.  Its first 
assertion is that, if we work modulo pi, the relations between the
geodetic angles "split" according to the squareclasses of their tangents.

In other words, if some integral linear combination,  say

               aA + bB + cC + ...       is a rational multiple of pi,

where  A,B,C,... are various geodetic angles,  and  a,b,c,... integers,
then so is the value of that same sum restricted to the  A  whose tangents
are in a given squareclass.

    This reduces us to finding the analogs of the Stormer theory for each
squareclass, the Stormer theory being that for squareclass 1.  

   These theories are all much the same, being simplest in the case when
the appropriate quaratic number field has unique factorization.  For 
instance the "Eisenstein" case,  when the tangents are rational multiples
of root3, reduces to the theory of Eisenstein integers.  

   But the lack of unique factorization doesn't make too much difference.
What happens in each case is that each prime  p  that splits in the number
field  Q(root(-d)) gives rise to an angle I'll call p_d,  which is  1/s 
times the argument of  a + b.root(-d), where  a,b  are (if I remember 
correctly) the smallest solution of  a^2 + d.b^2 = 4.p^s.   [In the UF
case we always have  s = 1; in general  s  divides the class number.]

    The argument of a general number  A + B.root(-d),  if  A  and  B
are coprime, is then a signed sum 

            +- a(p_d) +- b(q_d) +- c(r_d) +- ...    modulo pi

where  A^2 + d B^2 =  p^a.q^b.r^c...  , where  "modulo pi"  means
"modulo ratioonal multiples of  pi" - however, there's a bound
for the denominator - it must divide  h.(size of unit group)  I think.
In practice, it's always so small that there's no problem.


   For instance in the case  d = 1,  17+6i  has norm  325 = 5^2.13,
so  arctan(6/17) = +-2(5_1) +-(13_1)  modulo  pi,
where  5_1  means  arctan(1/2)  and  13_1  means  arctan(2/3).

   The "Stormer numbers"  are just another base, diagonally
related to this more natural one.
 
     The "hyperbolic" case is much the same except that the presence
of units makes there be, as well as the hyprbolic distances  p_d
for all primes  p,  a hyperbolic distance  1_d, and the modulus  pi
above gets replaced by  this very special distance.

   The base you call "hyperbolic Stormer numbers" is the base for
d = 1  obtained by the same type transformation as relates the
ordinary Stormer numbers to the Gaussian prime base.


  REgards,  John Conway