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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... An irreducible fraction is a fraction a/b, where the numerator a is an integer and the denominator b is a positive integer, such that there is not another fraction c/d with c smaller in absolute value than a and 0<d<b, and c and d are integers... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...

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. In mathematics, a series is often represented as the sum of a sequence of terms. ...

Examples

The Farey sequences of orders 1 to 8 are :

F₁ = {⁰⁄₁, ¹⁄₁}

F₂ = {⁰⁄₁, ¹⁄₂, ¹⁄₁}

F₃ = {⁰⁄₁, ¹⁄₃, ¹⁄₂, ²⁄₃, ¹⁄₁}

F₄ = {⁰⁄₁, ¹⁄₄, ¹⁄₃, ¹⁄₂, ²⁄₃, ³⁄₄, ¹⁄₁}

F₅ = {⁰⁄₁, ¹⁄₅, ¹⁄₄, ¹⁄₃, ²⁄₅, ¹⁄₂, ³⁄₅, ²⁄₃, ³⁄₄, ⁴⁄₅, ¹⁄₁}

F₆ = {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}

F₇ = {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}

F₈ = {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^[1]

... 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^[2]

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. the are cool The Geologist by Carl Spitzweg A geologist is a contributor to the science of geology, studying the physical structure and processes of the Earth and planets of the solar system (see planetary geology). ... John Farey, Sr. ... The Philosophical Magazine is arguably the world’s oldest commercially published scientific journal. ... 1816 was a leap year starting on Monday (see link for calendar). ... The Generalized Mediant also called Rational Mean and its very special case the Arithmonic Mean can be used to create new simple arithmetical roots-solving algorithms as well as re-discovering by means of the most simple arithmetic the well-known Bernoullis, Newtons, halleys and Householders... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 &#8211; May 23, 1857) was a French mathematician. ... --69. ...

Properties

Sequence 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 F₆ consists of F₅ 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) :- The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...

Using the fact that |F₁| = 2, we can derive an expression for the length of F_n :-

The asymptotic behaviour of |F_n| is :-

Farey neighbours

Fractions which are neighbouring terms in a Farey sequence 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

,

this is equivalent to saying that

bc − ad = 1.

Thus ¹⁄₃ and ²⁄₅ are neighbours in F₅, 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 — in other words, The Generalized Mediant also called Rational Mean and its very special case the Arithmonic Mean can be used to create new simple arithmetical roots-solving algorithms as well as re-discovering by means of the most simple arithmetic the well-known Bernoullis, Newtons, halleys and Householders...

.

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

,

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 F₈.

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. In number theory, the Stern-Brocot tree is a method of listing all non-negative rational numbers as well as a point representing infinity (here represented formally as 1/0). ...

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 In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

[0; a₁, a₂, …, a_{n − 1}, a_n, 1]

[0; a₁, a₂, …, 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; a₁, a₂, …, a_n]

and its other neighbour has a continued fraction expansion

[0; a₁, a₂, …, 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. In mathematics a Ford circle is a circle with centre at (p/q, 1/2q2) and radius 1/(2q2), where p/q is a fraction in its lowest terms (i. ...

For every fraction p/q (in its lowest terms) there is a Ford circle C[p/q], which is the circle with radius 1/(2q²) and centre at (p/q, 1/(2q²)). 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. In mathematics, two sets are said to be disjoint if they have no element in common. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

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 . 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 for any r>1/2, and that for any r>-1, are equivalent to the Riemann hypothesis. Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... Edmund Georg Hermann (Yehezkel) Landau (February 14, 1877 – February 19, 1938) was a German Jew mathematician and author of over 250 papers on number theory. ...

Simple Algorithm

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

 100 'UBASIC code to generate a Farey Sequence of order N in traditional order 110 N=7:NumTerms=1 120 A=0:B=1:C=1:D=N 140 print A;B 150 while (C<N) 160 NumTerms=NumTerms+1 170 K=int((N+B)/D) 180 E=K*C-A:F=K*D-B 190 A=C:B=D:C=E:D=F:print A;B 200 wend 210 print NumTerms 220 end 

(Note that no special handling is required near line 180 to reduce each term to the form of a reduced fraction)

To generate the sequence in descending order (non-traditional) substitute these lines:

 120 A=1:B=1:C=N-1:D=N 150 while (A>0) 

Brute force searches for solutions to Diophantine equations in rationals can often take advantage of the Farey series (to search only reduced forms); line 120 can also be modified to include any two adjacent terms so as to generate terms only larger (or smaller) than a given term. In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ...

References

1. ^ Hardy, G.H. & Wright, E.M. (1979) An Introduction to the Theory of Numbers (Fifth Edition). Oxford University Press. ISBN 0-19-853171-0
2. ^ Beiler, Albert H. (1964) Recreations in the Theory of Numbers (Second Edition). Dover. ISBN 0-486-21096-0

External links

• Farey series at cut-the-knot
• Stern-Brocot Tree at cut-the-knot
• MathWorld page on Stern-Brocot trees
• The Minkowski Question Mark and the Modular Group SL(2,Z) reviews the isomorphisms of the Stern-Brocot Tree.
• Symmetries of Period-Doubling Maps reviews connections between Farey Fractions and Fractals.