2020 IB Extended Essays

3.2.3. Observations of the Magic Table for √ and the Solutions to 2 − 2 = 1 From the above Magic Table for √ 7 and Magic Table for √ 13 , the following observations can be made of the nature of the pattern and periodicity manifest in the values of � � 2 and 2 − 2 = 1 . Observation 1 : The values of � � 2 bounce back and forth around and approximate more closely as the table continues. This can be explained by the alternative expression of the Pell’s equation: � � 2 = 2 +1 2 ( ∵ 2 = 2 + 1) . Since the value of increases infinitely as the table continues, it becomes evident that � � 2 should approximate , as lim →∞ � � 2 = lim →∞ 2 +1 2 = . Observation 2 : Based on Observation 1 , it is inevitable that should approximate √ . is referred to as the th convergent of √ . Observation 3 : The value of 2 − 2 can be deduced from the numerator of the mixed fraction expression of � � 2 ; e.g., 12 11877 11881 indicates that − 4 ≡ 11877 ( 11881) is going to be equal to 2 − 13 2 , because 12 11877 11881 = 13 + −4 11881 . Observation 4 : The statement made earlier about how the solutions to Pell’s equation 2 − 2 = 1 can be found from the palindromic portion { 1 , 2 , 3 , ⋯ , } of the periodic continued fraction notation of √ is confirmed. For example, the solutions to 2 − 7 2 = 1 can be found in ��√ � , 1 , 2 , 1 ������ , 2 �√ � ���������������������� � = � 2, 1, 1, 1 ��� , 4 ���������������� � , and the solutions to 2 − 13 2 = 1 can be found in ��√ � , 1 , 2 , 2 , 1 ��������� , 2 �√ � ������������������������� � = � 3, 1, 1, 1, 1 ����� , 6 ���������������� � . (In the

13