[seqfan] Arc-Tangent Irreducible Rationals

[pauldhanna at juno.com]

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? 

Thus, if h(x)=(x+1)/(x-1), then h(3)/h(2)=h(5), 
so 5 is not a hyperbolic Stormer number.

If we permit 1 to be a hyperbolic Stormer number (though a singularity), the sequence would begin 1,2,3,...

If non-trivial, what is the rest of the sequence of 'hyperbolic Stormer numbers'?  

To really stretch the generalization, we might consider 
the 'ELLIPTIC Stormer numbers' with some modulus k.
But we may save that idea for a very rainy day.

Thank you once again,
     Paul


>
>   A few years ago, I worked out the much more general theory of
>"geodetic angles" with L. Sadun and C.Radin.  These are the angles
>whose squared trigonometric functions are rational, and the theory
>tells you exactly all the rational linear relations between them.
>We gave a basis for the entire set that's analogous to Stormer's
>basis for those whose tangents are rational.
>
>    John Conway