Difference between revisions of "Farey series"
From Encyclopedia of Mathematics
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| − | ''of order | + | {{TEX|done}} |
| + | ''of order $n$'' | ||
| − | The increasing sequence of non-negative irreducible fractions not exceeding 1 with denominators not exceeding | + | 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. | The following assertions hold. | ||
| − | 1) If | + | 1) If $a/b$ and $a'/b'$ are two consecutive terms of the Farey series of order $n$, then |
| − | + | $$ba'-ab'=1.$$ | |
| − | 2) If | + | 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 | + | 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). | Farey series were investigated by J. Farey (1816). | ||
| Line 27: | Line 28: | ||
====Comments==== | ====Comments==== | ||
| − | Of course, in | + | 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
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