2020 IB Extended Essays

second example, however, the solution was found in the second repetition of the repeating portion.) Observation 5 : The values of 2 − 2 are eventually periodic. This is supported by a theorem, which will be assumed not be proven in this essay, for the sake of brevity and focus; it stems from the eventual periodicity of the continued fraction expansion of a quadratic irrational (www.math.uci.edu, [unknown]). Observation 6 : Because the values of 2 − 2 repeat periodically infinitely many times and there’s a solution to 2 − 2 = 1 in every period, there are infinitely many solutions to any given 2 − 2 = 1 . 3.2.4 Generating All Solutions to Pell’s Equation from the Fundamental Solution All solutions to a given Pell’s equation can be found by calculating every convergent and checking each, but there is a more straightforward method for generating new solutions from the known, fundamental solution ([unknown], www.math.uci.edu): 1. First, observe that ( 1 , 1 ) is the smallest, fundamental solution to the Pell’s equation 2 − 2 = 1 (smallest and among all positive solutions). 2. Suppose ( 2 , 2 ) is another solution to the equation; 2 2 − 2 2 = 1 . Then, consider ( 1 + 1 √ ) � 2 + 2 √ � = ( 1 2 + 1 2 ) + ( 1 2 + 1 2 ) √ , which then computes, ( 2 1 + 2 1 ) 2 − ( 2 1 + 2 1 ) 2 = 1 2 2 2 + 2 1 2 1 2 + 2 1 2 2 2 − 1 2 2 2 − 2 1 2 1 2 − 2 2 1 2 = 2 2 ( 1 2 − 1 2 ) − 2 2 ( 1 2 − 1 2 ) = 2 2 − 2 2 = 1 3. Therefore, ( 1 2 + 1 2 , 1 2 + 1 2 ) is another solution;

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