### Case 3

Next we study the case that and is odd.

In this subsection we use methods that are very similar to the one used in Subsection 5.4.2 and Subsection 5.4.3, and hence we omit the detail of the argument.
In Fig. 5.21 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 .
We suppose that

 (5.51)

and

 (5.52)

We also assume that

 and (5.53)

If we are to prove that , by Lemma 5.1 we have only to prove that is the smallest number that does not belong to .
.
We have .
. Therefore we have .
The case that can be treated in the similar way, so we omit it.