2020 IB Extended Essays

√ 13 = 3 + 1 1 �√ 13 − 3 � = 3 + 1 √ 13 + 3 4 = 3 + 1 1 + 1 4 √ 13 − 1

= 3 + 1 1 + 1 √ 13 + 1 3

= 3 + 1 1 + 1 1 + 1 3

= 3 + 1 1 + 1 1 + 1 √ 13 + 2 3

√ 13 − 2

= 3 + 1 1 + 1 1 + 1 1 + 1 3 √ 13 − 1 = 3 + 1 1 + 1 1 + 1 1 + 1 1 + 1 4

= 3 + 1 1 + 1 1 + 1 1 + 1 √ 13 + 1 4

1 1 + 1

= 3 +

1 + 1 1 + 1 1 + 1 (

√ 13 + 3)

√ 13 − 3

1 1 + 1 1 1 + 1

= 3 + = 3 +

1 + 1 1 + 1 1 + 1 6 + (

√ 13 − 3)

= ⋯

1 + 1 1 + 1 1 + 1 6 + 1 1 + 1 √ 13 + 1 3

These continued fraction expansions are evidently periodic and can be rewritten in continued fraction notations (in which the quotients of the expansions are listed in order) as √ 7 = [2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, ⋯ ] = [2, 1, 1, 1, 4 ���������� ] √ 13 = [3, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ⋯ ] = [3, 1, 1, 1, 1, 6 ������������� ]

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