| Farey Sequence |
Website Links For Sequence |
Information AboutFarey Sequence |
| CATEGORIES ABOUT FAREY SEQUENCE | |
| fractions | |
| number theory | |
| sequences | |
|
Each Farey sequence starts with the value 0, denoted by the fraction 0⁄1, and ends with the value 1, denoted by the fraction 1⁄1 (although some authors omit these terms). A Farey sequence is sometimes called a Farey Series , which is not strictly correct, because the terms are not summed. EXAMPLES The Farey sequences of orders 1 to 8 are : F F F F F F F F HISTORY The history of 'Farey series' is very curious ... once again the man whose name was given to a mathematical relation was not the original discoverer so far as the records go. Farey sequences are named after the British Geologist John Farey, Sr. , whose letter about these sequences was published in the '' Philosophical Magazine '' in 1816 . Farey conjectured that each term in a Farey sequence is the Mediant of its neighbours — however, so far as is known, he did not prove this property. Farey's letter was read by Cauchy , who provided a proof in his ''Exercises de mathématique'', and attributed this result to Farey. In fact, another mathematician, C. Haros , had published similar results in 1802 which were almost certainly not known either to Farey or to Cauchy. Thus it was a historical accident that linked Farey's name with these sequences. PROPERTIES Sequence length The Farey sequence of order ''n'' contains all of the members of the Farey sequences of lower orders. In particular ''Fn'' contains all of the members of ''F''''n''−1, and also contains an additional fraction for each number that is less than ''n'' and coprime to ''n''. Thus ''F''6 consists of ''F''5 together with the fractions 1⁄6 and 5⁄6. The middle term of a Farey sequence ''F''''n'' is always 1⁄2, for ''n'' > 1. From this, we can relate the lengths of ''Fn'' and ''F''''n''−1 using Euler's Totient Function φ(''n'') :- | ||
| Using The Fact That ''F''<sub>1</sub> | 2, we can derive an expression for the length of ''F<sub>n</sub>'' :- |
|
| :<math>F N | 1 + \sum_{m=1}^n arphi(m)</math> |
|
|