Farey sequence

In mathematics, the Farey sequence of order "n" is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to "n", arranged in order of increasing size.

Each Farey sequence starts with the value 0, denoted by the fraction ⁰⁄₁, and ends with the value 1, denoted by the fraction ¹⁄₁ (although some authors omit these terms).

A Farey sequence is sometimes called a Farey series, which is not strictly correct, because the terms are not summed.

Examples

The Farey sequences of orders 1 to 8 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}:"F"₅ = {⁰⁄₁, ¹⁄₅, ¹⁄₄, ¹⁄₃, ²⁄₅, ¹⁄₂, ³⁄₅, ²⁄₃, ³⁄₄, ⁴⁄₅, ¹⁄₁}:"F"₆ = {⁰⁄₁, ¹⁄₆, ¹⁄₅, ¹⁄₄, ¹⁄₃, ²⁄₅, ¹⁄₂, ³⁄₅, ²⁄₃, ³⁄₄, ⁴⁄₅, ⁵⁄₆, ¹⁄₁}:"F"₇ = {⁰⁄₁, ¹⁄₇, ¹⁄₆, ¹⁄₅, ¹⁄₄, ²⁄₇, ¹⁄₃, ²⁄₅, ³⁄₇, ¹⁄₂, ⁴⁄₇, ³⁄₅, ²⁄₃, ⁵⁄₇, ³⁄₄, ⁴⁄₅, ⁵⁄₆, ⁶⁄₇, ¹⁄₁}:"F"₈ = {⁰⁄₁, ¹⁄₈, ¹⁄₇, ¹⁄₆, ¹⁄₅, ¹⁄₄, ²⁄₇, ¹⁄₃, ³⁄₈, ²⁄₅, ³⁄₇, ¹⁄₂, ⁴⁄₇, ³⁄₅, ⁵⁄₈, ²⁄₃, ⁵⁄₇, ³⁄₄, ⁴⁄₅, ⁵⁄₆, ⁶⁄₇, ⁷⁄₈, ¹⁄₁}

History

:"The history of 'Farey series' is very curious" — Hardy & Wright (1979) Chapter III [Hardy, G.H. & Wright, E.M. (1979) "An Introduction to the Theory of Numbers" (Fifth Edition). Oxford University Press. ISBN 0-19-853171-0]

:"... once again the man whose name was given to a mathematical relation was not the original discoverer so far as the records go." — Beiler (1964) Chapter XVI [Beiler, Albert H. (1964) "Recreations in the Theory of Numbers" (Second Edition). Dover. ISBN 0-486-21096-0]

Farey sequences are named after the British geologist John Farey, Sr., whose letter about these sequences was published in the "Philosophical Magazine" in 1816. Farey conjectured that each new term in a Farey sequence expansion is the mediant of its neighbours — however, so far as is known, he did not prove this property. Farey's letter was read by Cauchy, who provided a proof in his "Exercises de mathématique", and attributed this result to Farey. In fact, another mathematician, C. Haros, had published similar results in 1802 which were almost certainly not known either to Farey or to Cauchy. Thus it was a historical accident that linked Farey's name with these sequences.

Properties

equence length

The Farey sequence of order "n" contains all of the members of the Farey sequences of lower orders. In particular "F_n" contains all of the members of "F"_"n"−1, and also contains an additional fraction for each number that is less than "n" and coprime to "n". Thus "F"₆ consists of "F"₅ together with the fractions ¹⁄₆ and ⁵⁄₆. The middle term of a Farey sequence "F"_"n" is always ¹⁄₂, for "n" > 1.

From this, we can relate the lengths of "F_n" and "F"_"n"−1 using Euler's totient function φ("n") :-

:$|F\_n|\; =\; |F\_\{n-1\}|\; +\; varphi(n)$

Using the fact that |"F"₁| = 2, we can derive an expression for the length of "F_n" :-

:$|F\_n|\; =\; 1\; +\; sum\_\{m=1\}^n\; varphi(m)$

The asymptotic behaviour of |"F_n"| is :-

:$|F\_n|\; sim\; frac\; \{3n^2\}\{pi^2\}$

Farey neighbours

Fractions which are neighbouring terms in any Farey sequence are known as a Farey pair and have the following properties.

If ^"a"⁄_"b" and ^"c"⁄_"d" are neighbours in a Farey sequence, with ^"a"⁄_"b" < ^"c"⁄_"d", then their difference ^"c"⁄_"d" − ^"a"⁄_"b" is equal to ¹⁄_"bd". Since

:$frac\{c\}\{d\}\; -\; frac\{a\}\{b\}\; =\; frac\{bc\; -\; ad\}\{bd\}$,

this is equivalent to saying that

:"bc" − "ad" = 1.

Thus ¹⁄₃ and ²⁄₅ are neighbours in "F"₅, and their difference is ¹⁄₁₅.

The converse is also true. If

:"bc" − "ad" = 1

for positive integers "a","b","c" and "d" with "a" < "b" and "c" < "d" then ^"a"⁄_"b" and ^"c"⁄_"d" will be neighbours in the Farey sequence of order max("b,d").

If ^"p"⁄_"q" has neighbours ^"a"⁄_"b" and ^"c"⁄_"d" in some Farey sequence, with

:^"a"⁄_"b" < ^"p"⁄_"q" < ^"c"⁄_"d"

then ^"p"⁄_"q" is the mediant of ^"a"⁄_"b" and ^"c"⁄_"d" &mdash; in other words,

:$frac\{p\}\{q\}\; =\; frac\{a\; +\; c\}\{b\; +\; d\}$.

And if ^"a"⁄_"b" and ^"c"⁄_"d" are neighbours in a Farey sequence then the first term that appears between them as the order of the Farey sequence is increased is

:$frac\{a+c\}\{b+d\}$,

which first appears in the Farey sequence of order "b" + "d".

Thus the first term to appear between ¹⁄₃ and ²⁄₅ is ³⁄₈, which appears in "F"₈.

The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 (= ⁰⁄₁) and 1 (= ¹⁄₁), by taking successive mediants.

Fractions that appear as neighbours in a Farey sequence have closely related continued fraction expansions. Every fraction has two continued fraction expansions - in one the final term is 1; in the other the final term is greater than 1. If ^"p"⁄_"q", which first appears in Farey sequence "F_q", has continued fraction expansions

: [0; "a"₁, "a"₂, …, "a"_{"n" − 1}, "a"_"n", 1] : [0; "a"₁, "a"₂, …, "a"_{"n" − 1}, "a"_"n" + 1]

then the nearest neighbour of ^"p"⁄_"q" in "F_q" (which will be its neighbour with the larger denominator) has a continued fraction expansion

: [0; "a"₁, "a"₂, …, "a"_"n"]

and its other neighbour has a continued fraction expansion

: [0; "a"₁, "a"₂, …, "a"_{"n" − 1}]

Thus ³⁄₈ has the two continued fraction expansions [0; 2, 1, 1, 1] and [0; 2, 1, 2] , and its neighbours in "F₈" are ²⁄₅, which can be expanded as [0; 2, 1, 1] ; and ¹⁄₃, which can be expanded as [0; 2, 1] .

Ford circles

There is an interesting connection between Farey sequence and Ford circles.

For every fraction "p"/"q" (in its lowest terms) there is a Ford circle C ["p"/"q"] , which is the circle with radius 1/(2"q"²) and centre at ("p"/"q", 1/(2"q"²)). Two Ford circles for different fractions are either disjoint or they are tangent to one another - two Ford circles never intersect. If 0 < "p"/"q" < 1 then the Ford circles that are tangent to C ["p"/"q"] are precisely the Ford circles for fractions that are neighbours of "p"/"q" in some Farey sequence.

Thus C [2/5] is tangent to C [1/2] , C [1/3] , C [3/7] , C [3/8] etc.

Riemann Hypothesis

Farey sequences are used in two equivalent formulations of the Riemann hypothesis. Suppose the terms of $F\_n$ are $\{a\_\{k,n\}\; :\; k\; =\; 0,\; 1,\; ldots\; m\_n\}$. Define $d\_\{k,n\}\; =\; a\_\{k,n\}\; -\; k/m\_n$, in other words $d\_\{k,n\}$ is the difference between the kth term of the nth Farey sequence, and the kth member of a set of the same number of points, distributed evenly on the unit interval. Franel and Landau proved that the two statements that $sum\_\{k=1\}^\{m\_n\}\; |d\_\{k,n\}|\; =\; mathcal\{O\}\; (n^r)$ for any r>1/2, and that $sum\_\{k=1\}^\{m\_n\}\; d\_\{k,n\}^2\; =\; mathcal\{O\}(n^r)$ for any r>-1, are equivalent to the Riemann hypothesis.

imple algorithm

A surprisingly simple algorithm exists to generate the terms in either traditional order (ascending) or non-traditional order (descending):

"""Python function to print the nth Farey sequence, either ascending or descending."""def farey( n, asc=True ): if asc: a, b, c, d = 0, 1, 1 , n # (*) else: a, b, c, d = 1, 1, n-1 , n # (*) print "%d/%d" % (a,b) while (asc and c < n) or (not asc and a > 0): k = int((n + b)/d) a, b, c, d = c, d, k*c - a, k*d - b print "%d/%d" % (a,b)

Brute force searches for solutions to Diophantine equations in rationals can often take advantage of the Farey series (to search only reduced forms). The lines marked (*) can also be modified to include any two adjacent terms so as to generate terms only larger (or smaller) than a given term. [Norman Routledge, "Computing Farey Series," The Mathematical Gazette, Vol. 29 (No. 523), 55-62 (March 2008).]

References

Footnotes

Further reading

* Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, "Concrete Mathematics: A Foundation for Computer Science", 2nd Edition (Addison-Wesley, Boston, 1989); in particular, Sec. 4.5 (pp. 115-123), Bonus Problem 4.61 (pp. 150, 523-524), Sec. 4.9 (pp. 133-139), Sec. 9.3, Problem 9.3.6 (pp. 462-463). ISBN 0201558025.
* Linas Vepstas. "The Minkowski Question Mark, GL(2,Z), and the Modular Group." http://linas.org/math/chap-minkowski.pdf reviews the isomorphisms of the Stern-Brocot Tree.
* Linas Vepstas. "Symmetries of Period-Doubling Maps." http://linas.org/math/chap-takagi.pdf reviews connections between Farey Fractions and Fractals.

External links

* Alexander Bogomolny. [http://www.cut-the-knot.org/blue/Farey.shtml Farey series] and [http://www.cut-the-knot.org/blue/Stern.shtml Stern-Brocot Tree] at http://www.cut-the-knot.org/
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