Farey sequence

Farey sequence Farey diagram to F9 represented with circular arcs. In the SVG image, hover over a curve to highlight it and its terms. Farey diagram to F9. Symmetrical pattern made by the denominators of the Farey sequence, F9. Symmetrical pattern made by the denominators of the Farey sequence, F25.

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction,[a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

With the restricted definition, 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.[2] Contents 1 Examples 2 History 3 Properties 3.1 Sequence length and index of a fraction 3.2 Farey neighbours 3.3 Farey neighbours and continued fractions 3.4 Farey fractions and the least common multiple 3.5 Farey fractions and the greatest common divisor 3.6 Applications 3.7 Ford circles 3.8 Riemann hypothesis 3.9 Other sums involving Farey fractions 4 Next term 5 See also 6 Footnotes 7 References 8 Further reading 9 External links 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 } Centered 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 } Sorted 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} iterations 1-10 Plotting the numerators versus the denominators of a Farey sequence gives a shape like the one to the right, shown for F6.

Reflecting this shape around the diagonal and main axes generates the Farey sunburst, shown below. The Farey sunburst of order n connects the visible integer grid points from the origin in the square of side 2n, centered at the origin. Using Pick's theorem the area of the sunburst is 4(|Fn|−1), where |Fn| is the number of fractions in Fn.

History The history of 'Farey series' is very curious — Hardy & Wright (1979)[3] ... 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)[4] 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, without offering proof, that each new term in a Farey sequence expansion is the mediant of its neighbours. Farey's letter was read by Cauchy, who provided a proof in his Exercices de mathématique, and attributed this result to Farey. In fact, another mathematician, Charles Haros, had published similar results in 1802 which were not known either to Farey or to Cauchy.[4] Thus it was a historical accident that linked Farey's name with these sequences. This is an example of Stigler's law of eponymy.

Properties Sequence length and index of a fraction 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 Fn is always 1 / 2 , for n > 1. From this, we can relate the lengths of Fn and Fn−1 using Euler's totient function {displaystyle varphi (n)}  : {displaystyle |F_{n}|=|F_{n-1}|+varphi (n).} Using the fact that |F1| = 2, we can derive an expression for the length of Fn:[5] {displaystyle |F_{n}|=1+sum _{m=1}^{n}varphi (m)=1+Phi (n),} where {displaystyle Phi (n)} is the summatory totient.

We also have : {displaystyle |F_{n}|={frac {1}{2}}left(3+sum _{d=1}^{n}mu (d)leftlfloor {tfrac {n}{d}}rightrfloor ^{2}right),} and by a Möbius inversion formula : {displaystyle |F_{n}|={frac {1}{2}}(n+3)n-sum _{d=2}^{n}|F_{lfloor n/drfloor }|,} where µ(d) is the number-theoretic Möbius function, and {displaystyle lfloor {tfrac {n}{d}}rfloor } is the floor function.

The asymptotic behaviour of |Fn| is : {displaystyle |F_{n}|sim {frac {3n^{2}}{pi ^{2}}}.} The index {displaystyle I_{n}(a_{k,n})=k} of a fraction {displaystyle a_{k,n}} in the Farey sequence {displaystyle F_{n}={a_{k,n}:k=0,1,ldots ,m_{n}}} is simply the position that {displaystyle a_{k,n}} occupies in the sequence. This is of special relevance as it is used in an alternative formulation of the Riemann hypothesis, see below. Various useful properties follow: {displaystyle I_{n}(0/1)=0,} {displaystyle I_{n}(1/n)=1,} {displaystyle I_{n}(1/2)=(|F_{n}|-1)/2,} {displaystyle I_{n}(1/1)=|F_{n}|-1,} {displaystyle I_{n}(h/k)=|F_{n}|-1-I_{n}((k-h)/k).} The index of {displaystyle 1/k} where {displaystyle n/(i+1)-1} is equivalent to the Riemann hypothesis, and then Edmund Landau[19] remarked (just after Franel's paper) that the statement {displaystyle sum _{k=1}^{m_{n}}|d_{k,n}|=O(n^{r})quad forall r>1/2} is also equivalent to the Riemann hypothesis.

Other sums involving Farey fractions The sum of all Farey fractions of order n is half the number of elements: {displaystyle sum _{rin F_{n}}r={frac {1}{2}}|F_{n}|.} The sum of the denominators in the Farey sequence is twice the sum of the numerators and relates to Euler's totient function: {displaystyle sum _{a/bin F_{n}}b=2sum _{a/bin F_{n}}a=1+sum _{i=1}^{n}ivarphi (i),} which was conjectured by Harold L. Aaron in 1962 and demonstrated by Jean A. Blake in 1966. A one line proof of the Harold L. Aaron conjecture is as follows. The sum of the numerators is {displaystyle {displaystyle 1+sum _{2leq bleq n}sum _{(a,b)=1}a=1+sum _{2leq bleq n}b{frac {varphi (b)}{2}}}} . The sum of denominators is {displaystyle {displaystyle 2+sum _{2leq bleq n}sum _{(a,b)=1}b=2+sum _{2leq bleq n}bvarphi (b)}} . The quotient of the first sum by the second sum is {displaystyle {frac {1}{2}}} .

Let bj be the ordered denominators of Fn, then:[20] {displaystyle sum _{j=0}^{|F_{n}|-1}{frac {b_{j}}{b_{j+1}}}={frac {3|F_{n}|-4}{2}}} and {displaystyle sum _{j=0}^{|F_{n}|-1}{frac {1}{b_{j+1}b_{j}}}=1.} Let aj/bj the jth Farey fraction in Fn, then {displaystyle sum _{j=1}^{|F_{n}|-1}(a_{j-1}b_{j+1}-a_{j+1}b_{j-1})=sum _{j=1}^{|F_{n}|-1}{begin{Vmatrix}a_{j-1}&a_{j+1}\b_{j-1}&b_{j+1}end{Vmatrix}}=3(|F_{n}|-1)-2n-1,} which is demonstrated in.[21] Also according to this reference the term inside the sum can be expressed in many different ways: {displaystyle a_{j-1}b_{j+1}-a_{j+1}b_{j-1}={frac {b_{j-1}+b_{j+1}}{b_{j}}}={frac {a_{j-1}+a_{j+1}}{a_{j}}}=leftlfloor {frac {n+b_{j-1}}{b_{j}}}rightrfloor ,} obtaining thus many different sums over the Farey elements with same result. Using the symmetry around 1/2 the former sum can be limited to half of the sequence as {displaystyle sum _{j=1}^{lfloor |F_{n}|/2rfloor }(a_{j-1}b_{j+1}-a_{j+1}b_{j-1})=3(|F_{n}|-1)/2-n-lceil n/2rceil ,} The Mertens function can be expressed as a sum over Farey fractions as {displaystyle M(n)=-1+sum _{ain {mathcal {F}}_{n}}e^{2pi ia}}   where   {displaystyle {mathcal {F}}_{n}}   is the Farey sequence of order n.

This formula is used in the proof of the Franel–Landau theorem.[22] Next term A surprisingly simple algorithm exists to generate the terms of Fn in either traditional order (ascending) or non-traditional order (descending). The algorithm computes each successive entry in terms of the previous two entries using the mediant property given above. If a / b and c / d are the two given entries, and p / q is the unknown next entry, then c / d  =  a + p / b + q . Since c / d is in lowest terms, there must be an integer k such that kc = a + p and kd = b + q, giving p = kc − a and q = kd − b. If we consider p and q to be functions of k, then {displaystyle {frac {p(k)}{q(k)}}-{frac {c}{d}}={frac {cb-da}{d(kd-b)}}} so the larger k gets, the closer p / q gets to c / d .

To give the next term in the sequence k must be as large as possible, subject to kd − b ≤ n (as we are only considering numbers with denominators not greater than n), so k is the greatest integer ≤  n + b / d . Putting this value of k back into the equations for p and q gives {displaystyle p=leftlfloor {frac {n+b}{d}}rightrfloor c-a} {displaystyle q=leftlfloor {frac {n+b}{d}}rightrfloor d-b} This is implemented in Python as follows: def farey_sequence(n: int, descending: bool = False) -> None: """Print the n'th Farey sequence. Allow for either ascending or descending.""" (a, b, c, d) = (0, 1, 1, n) if descending: (a, c) = (1, n - 1) print("{0}/{1}".format(a, b)) while (c <= n and not descending) or (a > 0 and descending): k = (n + b) // d (a, b, c, d) = (c, d, k * c - a, k * d - b) print("{0}/{1}".format(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.[23] See also ABACABA pattern Stern–Brocot tree Euler's totient function Footnotes ^ “The sequence of all reduced fractions with denominators not exceeding n, listed in order of their size, is called the Farey sequence of order n.” With the comment: “This definition of the Farey sequences seems to be the most convenient. However, some authors prefer to restrict the fractions to the interval from 0 to 1.” — Niven & Zuckerman (1972)[1] References ^ Niven, Ivan M.; Zuckerman, Herbert S. (1972). An Introduction to the Theory of Numbers (Third ed.). John Wiley and Sons. Definition 6.1. ^ Guthery, Scott B. (2011). "1. The Mediant". A Motif of Mathematics: History and Application of the Mediant and the Farey Sequence. Boston: Docent Press. p. 7. ISBN 978-1-4538-1057-6. OCLC 1031694495. Retrieved 28 September 2020. ^ Hardy, G.H.; Wright, E.M. (1979). An Introduction to the Theory of Numbers (Fifth ed.). Oxford University Press. Chapter III. ISBN 0-19-853171-0. ^ Jump up to: a b Beiler, Albert H. (1964). Recreations in the Theory of Numbers (Second ed.). Dover. Chapter XVI. ISBN 0-486-21096-0. Cited in "Farey Series, A Story". Cut-the-Knot. ^ Sloane, N. J. A. (ed.). "Sequence A005728". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. ^ Jump up to: a b Tomas, Rogelio (January 2022). "Partial Franel sums" (PDF). Journal of Integer Sequences. 25 (1). ^ Austin, David (December 2008). "Trees, Teeth, and Time: The mathematics of clock making". American Mathematical Society. Rhode Island. Archived from the original on 4 February 2020. Retrieved 28 September 2020. ^ Martin, Greg (2009). "A product of Gamma function values at fractions with the same denominator". arXiv:0907.4384 [math.CA]. ^ Wehmeier, Stefan (2009). "The LCM(1,2,...,n) as a product of sine values sampled over the points in Farey sequences". arXiv:0909.1838 [math.CA]. ^ Tomas Garcia, Rogelio (August 2020). "Equalities between greatest common divisors involving three coprime pairs" (PDF). Notes on Number Theory and Discrete Mathematics. 26 (3). doi:10.7546/nntdm.2020.26.3.5-7. ^ "Farey Approximation". NRICH.maths.org. Archived from the original on 19 November 2018. Retrieved 18 November 2018. ^ Eliahou, Shalom (August 1993). "The 3x+1 problem: new lower bounds on nontrivial cycle lengths". Discrete Mathematics. 118 (1–3): 45–56. doi:10.1016/0012-365X(93)90052-U. ^ Zhenhua Li, A.; Harter, W.G. (2015). "Quantum Revivals of Morse Oscillators and Farey-Ford Geometry". Chem. Phys. Lett. 633: 208–213. arXiv:1308.4470. doi:10.1016/j.cplett.2015.05.035. S2CID 66213897. ^ Tomas, R. (2014). "From Farey sequences to resonance diagrams". Physical Review Special Topics - Accelerators and Beams. 17: 014001. doi:10.1103/PhysRevSTAB.17.014001. ^ Harabor, Daniel Damir; Grastien, Alban; Öz, Dindar; Aksakalli, Vural (26 May 2016). "Optimal Any-Angle Pathfinding In Practice". Journal of Artificial Intelligence Research. 56: 89–118. doi:10.1613/jair.5007. ^ Hew, Patrick Chisan (19 August 2017). "The Length of Shortest Vertex Paths in Binary Occupancy Grids Compared to Shortest r-Constrained Ones". Journal of Artificial Intelligence Research. 59: 543–563. doi:10.1613/jair.5442. ^ Tomas, Rogelio (2020). "Imperfections and corrections". arXiv:2006.10661 [physics]. ^ Franel, Jérôme (1924). "Les suites de Farey et le problème des nombres premiers". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse (in French): 198–201. ^ Landau, Edmund (1924). "Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse (in German): 202–206. ^ Kurt Girstmair; Girstmair, Kurt (2010). "Farey Sums and Dedekind Sums". The American Mathematical Monthly. 117 (1): 72–78. doi:10.4169/000298910X475005. JSTOR 10.4169/000298910X475005. S2CID 31933470. ^ Hall, R. R.; Shiu, P. (2003). "The Index of a Farey Sequence". Michigan Math. J. 51 (1): 209–223. doi:10.1307/mmj/1049832901. ^ Edwards, Harold M. (1974). "12.2 Miscellany. The Riemann Hypothesis and Farey Series". In Smith, Paul A.; Ellenberg, Samuel (eds.). Riemann's Zeta Function. Pure and Applied Mathematics. New York: Academic Press. pp. 263–267. ISBN 978-0-08-087373-2. OCLC 316553016. Retrieved 30 September 2020. ^ Routledge, Norman (March 2008). "Computing Farey series". The Mathematical Gazette. Vol. 92, no. 523. pp. 55–62. Further reading Hatcher, Allen. "Topology of Numbers". Mathematics. Ithaca, NY: Cornell U. Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1989). Concrete Mathematics: A foundation for computer science (2nd ed.). Boston, MA: Addison-Wesley. pp. 115–123, 133–139, 150, 462–463, 523–524. ISBN 0-201-55802-5. — in particular, see §4.5 (pp. 115–123), Bonus Problem 4.61 (pp. 150, 523–524), §4.9 (pp. 133–139), §9.3, Problem 9.3.6 (pp. 462–463). Vepstas, Linas. "The Minkowski Question Mark, GL(2,Z), and the Modular Group" (PDF). — reviews the isomorphisms of the Stern-Brocot Tree. Vepstas, Linas. "Symmetries of Period-Doubling Maps" (PDF). — reviews connections between Farey Fractions and Fractals. Cobeli, Cristian; Zaharescu, Alexandru (2003). "The Haros-Farey sequence at two hundred years. A survey". Acta Univ. Apulensis Math. Inform. (5): 1–38. "pp. 1–20" (PDF). Acta Univ. Apulensis. "pp. 21–38" (PDF). Acta Univ. Apulensis. Matveev, Andrey O. (2017). Farey Sequences: Duality and Maps Between Subsequences. Berlin, DE: De Gruyter. ISBN 978-3-11-054662-0. Errata + Haskell code External links Bogomolny, Alexander. "Farey series". Cut-the-Knot. Bogomolny, Alexander. "Stern-Brocot Tree". Cut-the-Knot. Pennestri, Ettore. "A Brocot table of base 120". "Farey series", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Stern-Brocot Tree". MathWorld. OEIS sequence A005728 (Number of fractions in Farey series of order n) OEIS sequence A006842 (Numerators of Farey series of order n) OEIS sequence A006843 (Denominators of Farey series of order n) Archived at Ghostarchive and the Wayback Machine: Bonahon, Francis. Funny Fractions and Ford Circles (video). Brady Haran. Retrieved 9 June 2015 – via YouTube. Authority control: National libraries Japan Categories: Fractions (mathematics)Number theorySequences and series