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Published in:
2009 | OriginalPaper | Chapter
Unconditional Verification of the Regulator and the Class Number
Authors : Michael J. Jacobson Jr., Hugh C. Williams
Published in: Solving the Pell Equation
Publisher: Springer New York
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We have seen that if ∈
Δ
is the fundamental unit of
$$\mathcal{O} = \left[1, \sqrt{D} \right]$$
, then
$$t + u\sqrt D = \left\{ \begin{array}{ll} \epsilon_\Delta \;\; {\rm when} \; N(\epsilon_\Delta) = 1 \\ \epsilon_\Delta^2 \;\; {\rm when}\; N(\epsilon_\Delta) = -1\, , \end{array}\right. $$
, where
t, u
is the fundamental solution of the Pell equation. We have also seen in Chapter 12 that if we have a sufficiently accurate approximation
R
Δ
′
to
R
Δ
= log ∈
Δ
, then we can compute a compact representation of ∈
Δ
, from which it is a simple matter to determine certain properties of
t
and
u
. This is of particular importance when ∈
Δ
is very large, which, as we have pointed out, is often the case when
Δ
is large.