where 
.
The associated matrix 
is symmetrical and we define it's signature
and it's "whole fingerprint" as:
.
This will be denoted as:
is the second binary digit of 
and
. Occasionally, 
will be used.
, we say that it belongs to a pattern
when
.
Since 
we must have 
and similarly for 
. The second part holds from
is efficient, and
not too cumbersome since characterizes the sequence 
.
For a given 
, not a perfect square, the pattern associated
with 
will be denoted by 
(first kind), and notations
3.10, 3.12 completed
by:
, the pattern associated with
will be denoted by 
(second kind), and following
notations will be used:
are of both kinds
(but not for the same 
).
, we have
is even, then 
is the
fundamental solution of 
. Otherwise, this solution
is associated with 
.
(over the 
possibilities) signatures that actually occur for a pattern. Moreover,
these signatures can be grouped into three classes that depend only
of the existence and the parity of the middle quotient of the pattern
:
and 
, then (using notations of definition 4.2)
we have: 
(and not in 
!).
By recurrence, a pattern of even length is obtained when starting
from the empty pattern, i.e. from 
and a pattern of odd length when starting from 
where 
is the middle quotient of the pattern. Starting from the
corresponding signatures (bold-faced in 4.2)
and using iteratively 4.3, we obtain the
three given classes.
,
one moves along radiuses when 
is even and along circles when
is odd (clockwise along the inner circle and counter-clockwise
along the outer).
fingerprints that actually occur. The action induced
by group them into three
transitive classes, depending on the existence and parity of the middle
quotient. The cardinal of the none, even and odd class are respectively
, 
and 
, leading to the following
rules (stated using notations 4.1):
 |
(4.4) |
are equalities in while the 
are only congruences.
such that equation 
defines a quadratic integer 
are exactly the positive elements
of the arithmetic progression:
take four values modulo 
, except from the signature 
where it takes only two values. According to the signature of ,
these values of 
are given by:
,
or 
, all
the quotients such that 
verifies additionally 
(and therefore such that has an associated ) are given
by 4.5 where 
has to be set according
to the last column of 4.6 (where 
is as in 4.1). 
, we have 
(Bezout). Substituting 
into equation 
,
i.e. into 
, we obtain 
and therefore 
since , 
are coprime and 
is a quadratic integer.
For the remaining results, a computer aided proof is the easiest.
We consider the first five binary digits of , i.e. 
and refer these 
, 
, 
(
) as
"the" digits. The signature fixes the value of 
,
determining the value of 
and therefore the exact
value of 
.
For each signature, equation 
can be iteratively
solved modulo the successive powers of 
, fixing at each step the
value of another digit. When using the usual tools of formal computing,
these values must be carefully expressed since 
is false
outside of ! It is therefore convenient to express a sum
in by 
in .
We iterate this process as long as the equation remains linear in
at least a variable. It happens that 
depends only
on 
and that the set 
collapses to the same set of cardinal 
for each 
-uple in the
remaining digits, except for signature
where it appears two sets of cardinal two : 
and 
.
The same formulae show that, for any first kind integer, 
depend only on 
(and not on the digit 
)
and give the value of 
leading to 
(when possible, i.e. for adequate signatures).
The patterns of first kind are characterized
by : the signature is not 
. The
middle element of a that allows 
is even
(or none). This last point is only an implication, not a characterization.
follows directly from theorem 4.10
by substituting 
into 4.5. By
definition, 
for an 
. Thus 
and the conclusion on 
follows. When is empty, we
have 
and conclusion remains when using 
instead of
.
-patterns).
(1) "nothing but 
", where all the quotients
are equal to a given .
(2) "any, one, one", where 
and 
(the sequence 
must be palindromic, since has to
be so).
. The corresponding
and 
are given by:
and 
is the -th
Fibonacci number. Additionally, the parameter 
must be
even when 
. 
. Thus the signatures are given by:
, is also a pattern
of first kind and the parity condition over is here to
select integers of second kind). The and
values follow from :
associated to a 
belong to the same pattern of first kind 
,
where 
. And we have
whose signature is or 
and whose quotients 
are all even and at least . Then
it exists infinitely many 
such that 
,
and all the associated 
share the above described pattern
of second kind .
according to :
pattern follows from :
is a fixed point of 
when is a fixed point of 
.
Conversely: if is as described, the existence of the 
such that 
follows from theorem 4.10.
Then proposition 4.13 shows that 
is a quadratic integer. And, by 4.8,
is a fixed point of .
It exists an infinite number of quadratic integers of each kind having a given period length.
given in proposition 4.15
and corollary 4.17 for a given value of 
or are quickly huge, and give no hint about the smallest
leading to a given period length.
. 
and 
. When 
is "nothing
but ones" and 
then 
is "nothing but fours" and 
.