Farey sequence

Farey sequence

[fair-ee]
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 01, and ends with the value 1, denoted by the fraction 11 (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 :
F1 = {01, 11}
F2 = {01, 12, 11}
F3 = {01, 13, 12, 23, 11}
F4 = {01, 14, 13, 12, 23, 34, 11}
F5 = {01, 15, 14, 13, 25, 12, 35, 23, 34, 45, 11}
F6 = {01, 16, 15, 14, 13, 25, 12, 35, 23, 34, 45, 56, 11}
F7 = {01, 17, 16, 15, 14, 27, 13, 25, 37, 12, 47, 35, 23, 57, 34, 45, 56, 67, 11}
F8 = {01, 18, 17, 16, 15, 14, 27, 13, 38, 25, 37, 12, 47, 35, 58, 23, 57, 34, 45, 56, 67, 78, 11}

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 discoverer so far as the records go. — Beiler (1964) Chapter XVI

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

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 16 and 56. The middle term of a Farey sequence Fn is always 12, for n > 1.

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

|F_n| = |F_{n-1}| + varphi(n)

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

|F_n| = 1 + sum_{m=1}^n varphi(m)

The asymptotic behaviour of |Fn| 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 ab and cd are neighbours in a Farey sequence, with ab < cd, then their difference cd − ab is equal to 1bd. Since

frac{c}{d} - frac{a}{b} = frac{bc - ad}{bd},

this is equivalent to saying that

bcad = 1.

Thus 13 and 25 are neighbours in F5, and their difference is 115.

The converse is also true. If

bcad = 1

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

If pq has neighbours ab and cd in some Farey sequence, with

ab < pq < cd

then pq is the mediant of ab and cd — in other words,

frac{p}{q} = frac{a + c}{b + d}.

And if ab and cd 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 13 and 25 is 38, which appears in F8.

The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 (= 01) and 1 (= 11), 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 pq, which first appears in Farey sequence Fq, has continued fraction expansions

[0; a1, a2, …, an − 1, an, 1]
[0; a1, a2, …, an − 1, an + 1]

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

[0; a1, a2, …, an]

and its other neighbour has a continued fraction expansion

[0; a1, a2, …, an − 1]

Thus 38 has the two continued fraction expansions [0; 2, 1, 1, 1] and [0; 2, 1, 2], and its neighbours in F8 are 25, which can be expanded as [0; 2, 1, 1]; and 13, 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.

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.

Simple 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.

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.

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