### 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 .