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Case 2
In this subsection we study the case that
and
is odd.
An example of this case is Example 5.4.
The argument here looks complicated, but it is a generalization of argument used in Example 5.4
.
In Fig. 5.20 we suppose that Predictions 5.1, 5.2, 5.3,5.4 and 5.5 are valid for
and for
. We also suppose that these predictions are valid for
and
. Under these assumptions we prove that
.
For
by Prediction 5.1 the first number from the right end of the row is
.
If we start with
and go leftward picking up every other number, then we have
which is an arithmetic sequence with common difference of
.
On the other hand if we start with
that is the second number from the right end of the row and go leftward picking up every other number, then we have an arithmetic sequence with common difference of
whose length is
, and hence the last number of this sequence is
. This number
is the first number of an arithmetic sequence with common difference of
and the last number of this sequence is
.
By Prediction 5.5 for
the number at the top of the column is
. If we move down picking up every other number, then we have an arithmetic sequence with common difference of
.
On the other hand if we start with 1 that is the second number of the column and move down picking up every other number, then we have
that starts as an arithmetic sequence with common difference of
such that it has
terms and the last number is
, then it becomes an arithmetic sequence with common difference of
whose first number is
.
In Fig. 5.20 we suppose that

(5.42) 

(5.43) 
and 
(5.44) 
By Lemma 5.1 we have only to prove that
is the smallest number that does not belong to
to get
.
By Definition 5.2
can be separated into three parts
,
and
.

(5.45) 

(5.46) 

(5.47) 

(5.48) 

(5.49) 

(5.50) 
By (5.45) and (5.46) we have
.
By (5.49) and (5.50) all the numbers in
are bigger than
.
By (5.47) and (5.48)
.
We know that the list
does not contain
, since the numbers in
the list and the number
have opposite parity.
In other words if
is odd, then the numbers in the list are even, and if
is even, then the numbers in the list are odd.
.
Therefore we have
.
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Up:Abstract and the table of contents Previous: Case 1