Ahmes expansions over certain Euclidean domains
It is well-known that if a and b are nonzero natural numbers then there exist nonzero natural numbers p0, p1,···, pn such that [special characters omitted], where p0 < p 1 < … < pn. Such expansions of [special characters omitted] are called "Ahmes expansions" of [special characters omitted]. This thesis is dedicated to extending this notion to the context of an arbitrary Euclidean domain. After a brief history and essential background on Ahmes expansions of positive rational numbers, the author advances an algorithm that can be utilized to produce Ahmes-type expansions of proper fractions over the integers and polynomial rings over a field, respectively. The author then develops several results related to the analogous concept of "length" of proper fractions for these specific Euclidean domains.