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Farey sequence

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

Each Farey sequence starts with the value 0, denominated by the fraction 0/1, and ends with the value 1, denominated 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 Fary 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

... once again the man whose name was given to a mathematical relation was not the original discover so far as the records go. -- Beiler (1964) Chapter XVI

Farey sequences are named after the
British geologist John Farey, whose letter about these sequences was published in the Philosophical Magazine in 1816. Farey conjectured that each term in a Farey sequence 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 is an historical accident that has linked Farey's name with these sequences.

Properties

Sequence length

The Farey sequence of order n contains all of the members of the Farey sequences of lower orders. In particular Fn contains all of the members of Fn-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 is always 1/2.

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

The asymptotic behaviour of |Fn| 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 c/d-a/b=(bc-ad)/bd, 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 min(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, p/q=(a+b)/(c+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 (a+b)/(c+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 F8.

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 Fq, has continued fraction expansions

then the nearest neighbour of p/q in Fq (which will be its neighbour with the larger denominator) has a continued fraction expansion

and its other neighbour has a continued fraction expansion

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 1/2q2 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.

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

References

• Beiler, Albert H. (1964) Recreations in the Theory of Numbers (Second Edition). Dover. ISBN 0486210960

• Hardy, G.H. & Wright, E.M. (1979) An Introduction to the Theory of Numbers (Fifth Edition). Oxford University Press. ISBN 0198531710