2020 IB Extended Essays

a new solution ( 1 2 + 1 2 , 1 2 + 1 2 ) can be generated from the fundamental solution, ( 1 , 1 ) . 4. Then, by induction, all ( , ) such that + √ = ( 1 + 1 √ ) , ∀ ∈ ℤ + are solutions to the Pell’s equation 2 − 2 = 1 . For example, consider the Pell’s equation 2 − 7 2 = 1 : The fundamental solution is ( 1 , 1 ) = (8, 3). Therefore, the solutions generated from it are 2 + 2 √ 7 = � 1 + 1 √ 7 � 2 = � 8 + 3 √ 7 � 2 = 127 + 48 √ 7 → ( 2 , 2 ) = (127, 48) 3 + 3 √ 7 = � 1 + 1 √ 7 � 3 = � 8 + 3 √ 7 � 3 = 2024 + 765 √ 7 → ( 3 , 3 ) = (2024, 765) ⋮ Consider the second example, the Pell’s equation 2 − 13 2 = 1 : The fundamental solution is ( 1 , 1 ) = (649, 180) . Therefore, the solutions generated from it are 2 + 2 √ 13 = � 649 + 180 √ 13 � 2 = 842401 + 233640 √ 13 ; ( 2 , 2 ) = (842401, 233640) , 3 + 3 √ 13 = � 649 + 180 √ 13 � 3 = 1093435849 + 303264540 √ 13 ; ( 3 , 3 ) = (1093435849, 303264540) , ⋮ This method and its examples confirm not only the Magic Tables, but also Observation 6 , that all Pell’s equations each have infinitely many solutions. The infinitely many positive integers will correspond to the infinitely many solutions ( , ) such that + √ = ( 1 + 1 √ ) .

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