2020 IB Extended Essays

As such, a square root of a square-free integer has a periodic continued fraction of the form √ = ��√ � , 1 , 2 , 3 , ⋯ , 3 , 2 , 1 ��������������� , 2 �√ � ������������������������������������� � (Junod, A 2015). That is, the repeating portion (excluding the last term, which is double the zeroth, non- periodic term) is a palindrome. And the repeating portion excluding the last term (i.e., { 1 , 2 , 3 , ⋯ , } ) is where the solutions to Pell’s equation can be found (Yang, SH 2008). Therefore, the continued fraction technique provides solutions to Pell’s equation, for which a solution always exists. 3.2.2 The “Magic Table” Algorithm The continued fraction notations from above can be linked back to Pell’s equation via a device called the “magic table”. The magic table operates by the following algorithm: 1. Draw a table and fill it in with 0 1 1 0 on the first two columns of the second and third row, and the numbers of the √ continued fraction expansion starting from the third column on the first row, as per the example below, Magic Table for √ 7 . Magic Table for √ 7 : 2 1 1 1 4 1 1 1 4 1 ⋯ 0 1 ⋯ 1 0

⋯ ⋯ ⋯

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