   Next: Two Families of Continued Up: Families of Continued Fractions Previous: The Fundamental Correspondence

# Some Propositions and Lemmas

Fractions" The following proposition is fundamental to the continued fractions of surds, because it can determine whether a set of partial quotients come from a given surd.

Proposition 1   Let ; then if and only if where , and is a fractional linear transformation. Since the partial quotients that come from a surd are almost palindromic, it is possible to produce more explicit forms of Proposition 1. The following lemmas reduce the amount of calculation necessary to verify that a set of partial quotients come from a surd by half. They will be used in the proof of the theorems in Section . The rest of the report contains new ideas and results, which are discussed in .

Lemma 1   Let D be a non-square integer, be the greatest integer in , , and . If such that , then  Lemma 2   Let D be a non-square integer, be the greatest integer in , , and . If such that , then  Now that these two lemmas have been stated, it is convenient to introduce and prove the theorems regarding two families of continued fractions.   Next: Two Families of Continued Up: Families of Continued Fractions Previous: The Fundamental Correspondence
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2000-01-06