  Shanks's sequence of quadratic fields $\Q(\sqrt{S_{n}})$ where
$S_{n}=(2^n+1)^2 + 2^{n+2}$ instances a class of quadratic fields for which the
class number is large and, therefore, the continued fraction period is
relatively short. Indeed, that period length increases linearly with $n$, that
is: in arithmetic progression. The fields have regulator $O(n^2)$. In the late
nineties, these matters intrigued Irving Kaplansky, and led him to compute
period length of the square root of sequences $a^2x^{2n}+bx^{n}+c$ for integers
$a$, $b$, $c$, and $x$. In brief, Kap found unsurprisingly that, generically,
triples $(a,b,c)$ are `leapers': they yield sequences with period length
increasing at exponential rate. But there are triples yielding sequences with
constant period length, Kap's `sleepers'. Finally, there are triples, as
exemplified by the Shanks's sequence, for which the period lengths increase in
arithmetic progression. Felicitously, Kaplansky called these `creepers'. It
seems that the sleepers and creepers are precisely those for which one is able
to detail the explicit continued fraction expansion for all $n$. Inter alia,
this thesis noticeably extends the known classes of creepers and finds that not
all are `kreepers' (of the shape identified by Kaplansky) and therefore not of
the shape of examples studied by earlier authors looking for families of
quadratic number fields with explicitly computable unit and of relatively large
regulator. The work of this thesis includes the discovery of old and new
families of hyperelliptic curves of increasing genus $g$ and torsion divisor of
order $O(g^2)$. It follows that the apparent trichotomy leaper/sleeper/creeper
coincides with the folk belief that the just-mentioned torsion is maximum
possible.
