A Farey sequence of order is the sequence of all completely reduced fractions between 0 and 1 where, when in lowest terms, each fraction has a denominator less than or equal to . Each fraction starts with 0, denoted by the fraction 0/1, and ends in 1, denoted by the fraction 1/1.
Farey sequences of orders 1-4 are:
Where denotes a Farey sequence of order .
Which is bigger, or ? Which is bigger, or ? Do you see a pattern?
A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, contains all the terms in . Also, contains an extra term for every number less than relatively prime to . Thus, we can write
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