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Difference between revisions of "Farey series"

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$$ba'-ab'=1.$$
 
$$ba'-ab'=1.$$
  
2) If $a/b$, $a'/b'$, $a''/b''$ are three consecutive terms of the Farey series of order $n$, 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]]
  
 
$$\frac{a''}{b''}=\frac{a+a'}{b+b'}.$$
 
$$\frac{a''}{b''}=\frac{a+a'}{b+b'}.$$
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3) The number of terms in the Farey series of order $n$ is equal to
 
3) The number of terms in the Farey series of order $n$ is equal to
  
$$1+\sum_{x=1}^n\phi(x).\tag{*}$$
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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 \ref{*} $\phi$ 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=32029
This article was adapted from an original article by V.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article