2020 IB Extended Essays

Magic Table for √ 7 : 2 1

1 ⋯ 717 ⋯

1

1

4

1

1

1

4

0

1

2

3

5

8

37

45

82

127

590

1 1 271 ⋯ 2 1 3 1 5 2 8 3 37 14 45 17 82 31 127 48 590 223 717 271 ⋯ � � 2 4 9 6 1 4 7 1 9 6 193 196 7 2 289 6 958 961 7 1 2304 6 49726 49729 7 2 73441 ⋯ 2 − 7 2 − 3 2 − 3 1 − 3 2 − 3 1 − 3 2 ⋯ 4. Identify the columns in which the Pell’s equation 2 − 7 2 = 1 is satisfied as per above. Determine the first of these columns and redefine the ( , ) of that column as ( 1 , 1 ) = (8, 3) , the fundamental solution to 2 − 7 2 = 1 ; the next ( , ) ’s that satisfy 2 − 7 2 = 1 are redefined as ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ), ⋯ and so on, in order of occurrence. Then, the ordered pairs of ( , ) correspond to all the solutions to the Pell’s equation 2 − 7 2 = 1 . From the continued fraction expansion of √ 13 , the Magic Table for √ 13 can likewise be filled out. Magic Table for √ 13 : 3 1 1 1 1 6 1 1 1 1 6 ⋯ 0 1 3 4 7 11 18 119 137 256 393 649 4287 ⋯ 1 0 1 1 2 3 5 33 38 71 109 180 1189 ⋯ 3 1 4 1 7 2 11 3 18 5 119 33 137 38 256 71 393 109 649 180 4287 1189 ⋯ � � 2 9 16 12 1 4 13 4 9 12 24 25 13 4 1089 12 1441 1444 13 3 5041 12 11877 11881 13 1 32400 12 1413717 1413721 ⋯ 2 − 13 2 − 4 3 − 3 4 − 1 4 − 3 3 − 4 1 − 4 ⋯ 1 2 3 14 17 31 48 223 0

12