Continued fractions with irrational numerators

Abstract

A continued fraction is called a simple continued fraction when the numerator is 1 and non-simple otherwise. Simple continued fractions of square roots have particularly "nice" periodic patterns. For example, sqrt{19} = [4,{2, 1, 3, 1, 2, 8}]_1 where the sequence 2, 1, 3, 1, 2, 8 is the periodic part which repeats. The pattern is "nice" because the first term is always half the last terms and the sequence 2, 1, 3, 1, 2 is identical when reversed. In this report, we study the periodic behavior of non-simple continued fractions of general square root with square root numerators. We show that these have many similarities to simple continued fraction expansions of square roots.