[seqfan] Arc-Tangent Irreducible Rationals

[John Conway]

On Fri, 6 Jun 2003, Paul D Hanna wrote:

> Apologize for the prior e-mail; after a little more tinkering, 
> I see that the reciprocal of the Stormer numbers do indeed 
> form the 'arc-tangent irreducible rationals'.  
> There are no others.  This was not as obvious to me 
> as it probably is to most of you.

   I'm not sure if you've read "The Book of Numbers"?  We have a 
section in there about these things (which is where the term
"Stormer number" was first defined).

> The Stormer numbers are now very interesting to me,
> because the entire set of arc-tangents of the rationals 
> can then be expressed as sums of +-arctan(1/A005528(n)).
> This implies that the Stormer numbers are related to 
> the Gaussian integers.

   Yes of course.  They are, for each prime p that's a sum of two
squares, the smallest  n  for which  n+i  has norm divisible by p.
Effectively, they "are" the "new" Gaussian primes, except that each
is multiplied up by earlier ones into the form  n+i.

   So Stormer's theorem (that the arc-cotangents of these numbers
form a basis for the space of arc-tangents of rationals) is really
just the unique factorization theorem for the Gaussian integers,
but modulo the reals.

> Does anyone know of a more direct formula for the
> n-th Stormer number than that given in A005528?

   I'm not sure what's given there.  They are simplest defined as
the numbers  n  for which the largest prime factor of  n^2 + 1
is at least  2n.

   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