Won't you be my i-Neighbor?
* In unit rank 1, there is a cyclic infrastructure.
* In unit rank 2, the infrastructure is biperiodic.
* Biperiodic - like a donut ? Yes!
* Currently only the baby steps are defined : ineighbors.
* If
,
then
* Since q ≡ 1 mod 3,
such that u3 = 1.
* Nontrivial embeddings:
is a minimum in a if
for
all i implies
for some
*
Define
for at least one 
* If
is minimal with respect to
≤ i,
α
is an i-neighbor of θ.
Finding Two Fundamental Units
* Beginning with OK and θ = 1, compute the
0-neighbor of θ, θ1. Compute it's 0-neighbor,
θ2, and so on. This is the 0-chain. Eventually
for some p and l,
*
If θ is a minimum in OK, then ( θ-1) is a reduced
principal ideal.
* Thus
is a fundamental unit of
OK.
*
Now find ø1, the 2-neighbor of θ, ø2 the
2-neighbor of ø1, etc. This is the 2-chain.
* Eventually
* Then
is the other fundamental
unit of OK.
*
Speeding this up requires defining giant steps:
efficient ideal multiplication or in general finding
faster ways to navigate the infrastructure.
Examples
Computing the fundamental units yields:
Preperiod Length: p = 0
First Period Length: l = 40
Second Period Length: m = 2
Regulator R = 196, which is less than lm = 80.
The first period begins:
More of the Periods
Second period:
Two Fundamental Units
The Donut
Well, it's more of a bumpy, twisted donut.
Here are 114 elements of the infrastructure:
