Difference between revisions of "Farey sequence"
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<math>F_4=\{0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1\}</math> | <math>F_4=\{0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1\}</math> | ||
| − | Where <math>F_n</math> denotes a | + | Where <math>F_n</math> denotes a Farey sequence of order <math>n</math>. |
| + | |||
| + | ==Proof Sketch== | ||
| + | Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+1}</math>? | ||
| + | |||
| + | Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+2}</math>? | ||
| + | |||
| + | Do you see a pattern? | ||
| + | |||
| + | Assume <math>a</math> and <math>b</math> are positive. | ||
==Properties== | ==Properties== | ||
| − | + | ===Sequence length=== | |
A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, <math>F_n</math> contains all the terms in <math>F_{n-1}</math>. Also, <math>F_n</math> contains an extra term for every number less than <math>n</math> relatively prime to <math>n</math>. Thus, we can write | A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, <math>F_n</math> contains all the terms in <math>F_{n-1}</math>. Also, <math>F_n</math> contains an extra term for every number less than <math>n</math> relatively prime to <math>n</math>. Thus, we can write | ||
| − | <math>#(F_n)=#(F_{n-1})+\phi{n}</math> | + | <math>\#(F_n)=\#(F_{n-1})+\phi{(n)}</math> |
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{{stub}} | {{stub}} | ||
Latest revision as of 03:16, 30 January 2021
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.
Examples
Farey sequences of orders 1-4 are:
Where 
denotes a Farey sequence of order .
Proof Sketch
Which is bigger, 
or 
?
Which is bigger, or 
?
Do you see a pattern?
Assume 
and 
are positive.
Properties
Sequence length
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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