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# Continued Fraction Expansions

Explores the nature of the continued fraction expansions of the square roots of the natural numbers less than or equal to 1 million.
by

## Matt Trapani

on 26 May 2010

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#### Transcript of Continued Fraction Expansions

As presented by Matt Trapani Continued Fraction Expansions of Goal Patterns Data I used to find answers Continued Fraction Expansion
Notation Answering 8 (a) Answer: n=1,680/2 = 840 which has 39 different expansions. Answering 8 (c) Which n has the most expansions of period 6
with the first two terms in the continued fraction expansion equal to 1? Answering 8 (b) What do we notice?
It looks like the greatest
expansion occurs at
0 (mod 5), 1 (mod 5),
2 (mod 5), 3 (mod 5),
4 (mod 5), and back to
0 (mod 5) again. Period 4 Expansion Table Questions or comments? looking for the n with the most period 4 expansions with the first term equal to 1 looking for the n with the most period 2 expansions Useful Patterns Table of Factors for Numbers Less Than 2,000
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