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''of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382302.png" />''
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{{TEX|done}}
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''of order $n$''
  
The increasing sequence of non-negative irreducible fractions not exceeding 1 with denominators not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382303.png" />. For example, the Farey series of order 5 is the sequence
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The increasing sequence of non-negative irreducible fractions not exceeding 1 with denominators not exceeding $n$. For example, the Farey series of order 5 is the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382304.png" /></td> </tr></table>
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$$\frac01,\frac15,\frac14,\frac13,\frac25,\frac12,\frac35,\frac23,\frac34,\frac45,\frac11.$$
  
 
The following assertions hold.
 
The following assertions hold.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382306.png" /> are two consecutive terms of the Farey series of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382307.png" />, then
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1) If $a/b$ and $a'/b'$ are two consecutive terms of the Farey series of order $n$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382308.png" /></td> </tr></table>
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$$ba'-ab'=1.$$
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f0382309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f03823010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f03823011.png" /> are three consecutive terms of the Farey series of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f03823012.png" />, then
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2) If $a/b$, $a'/b'$, $a''/b''$ are three consecutive terms of the Farey series of order $n$, then the [[mediant]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f03823013.png" /></td> </tr></table>
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$$\frac{a''}{b''}=\frac{a+a'}{b+b'}.$$
  
3) The number of terms in the Farey series of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f03823014.png" /> is equal to
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3) The number of terms in the Farey series of order $n$ is equal to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f03823015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$1+\sum_{x=1}^n\phi(x).\label{*}\tag{*}$$
  
 
Farey series were investigated by J. Farey (1816).
 
Farey series were investigated by J. Farey (1816).
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====Comments====
 
====Comments====
Of course, in (*) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038230/f03823016.png" /> denotes the [[Euler function|Euler function]].
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Of course, in \eqref{*} $\phi$ denotes the [[Euler function|Euler function]].

Latest revision as of 11:54, 2 January 2021

of order $n$

The increasing sequence of non-negative irreducible fractions not exceeding 1 with denominators not exceeding $n$. For example, the Farey series of order 5 is the sequence

$$\frac01,\frac15,\frac14,\frac13,\frac25,\frac12,\frac35,\frac23,\frac34,\frac45,\frac11.$$

The following assertions hold.

1) If $a/b$ and $a'/b'$ are two consecutive terms of the Farey series of order $n$, then

$$ba'-ab'=1.$$

2) If $a/b$, $a'/b'$, $a''/b''$ are three consecutive terms of the Farey series of order $n$, then the mediant

$$\frac{a''}{b''}=\frac{a+a'}{b+b'}.$$

3) The number of terms in the Farey series of order $n$ is equal to

$$1+\sum_{x=1}^n\phi(x).\label{*}\tag{*}$$

Farey series were investigated by J. Farey (1816).

References

[1] A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)
[2] R.R. Hall, "A note on Farey series" J. London Math. Soc. , 2 (1970) pp. 139–148
[3] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)


Comments

Of course, in \eqref{*} $\phi$ denotes the Euler function.

How to Cite This Entry:
Farey series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Farey_series&oldid=16556
This article was adapted from an original article by V.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article