Difference between revisions of "Farey sequence"

(Sequence length)
(Examples: Proof Sketch)
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Where <math>F_n</math> denotes a Farey sequence of order <math>n</math>.
 
Where <math>F_n</math> denotes a Farey sequence of order <math>n</math>.
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==Proof Sketch==
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Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+1}</math>?
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Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+2}</math>?
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Do you see a pattern?
  
 
==Properties==
 
==Properties==

Revision as of 04:12, 30 January 2021

A Farey sequence of order $n$ 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 $n$. 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:

$F_1=\{0/1, 1/1\}$

$F_2=\{0/1, 1/2, 1/1\}$

$F_3=\{0/1, 1/3, 1/2, 2/3, 1/1\}$

$F_4=\{0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1\}$

Where $F_n$ denotes a Farey sequence of order $n$.

Proof Sketch

Which is bigger, $\frac{a}{b}$ or $\frac{a+1}{b+1}$? Which is bigger, $\frac{a}{b}$ or $\frac{a+1}{b+2}$? Do you see a pattern?

Properties

Sequence length

A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, $F_n$ contains all the terms in $F_{n-1}$. Also, $F_n$ contains an extra term for every number less than $n$ relatively prime to $n$. Thus, we can write

$\#(F_n)=\#(F_{n-1})+\phi{(n)}$


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