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=> Algebraic Curves-Mordell Curve
=> Algebraic Curves-Ochoa Curve
=> Algebraic Integer
=> Algebraic Number
=> Algebraic Number Theory
=> Chebotarev Density Theorem
=> Class Field
=> Cyclotomic Field
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=> Casting Out Nines
=> A-Sequence
=> Anomalous Cancellation
=> Archimedes' Axiom
=> B2-Sequence
=> Calcus
=> Calkin-Wilf Tree
=> Egyptian Fraction
=> Egyptian Number
=> Erdős-Straus Conjecture
=> Erdős-Turán Conjecture
=> Eye of Horus Fraction
=> Farey Sequence
=> Ford Circle
=> Irreducible Fraction
=> Mediant
=> Minkowski's Question Mark Function
=> Pandigital Fraction
=> Reverse Polish Notation
=> Division by Zero
=> Infinite Product
=> Karatsuba Multiplication
=> Lattice Method
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=> Goodstein Sequence
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=> Nonregular Number
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=> One-Seventh Ellipse
=> Quaternary
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=> Repeating Decimal
=> Saunders Graphic
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=> Unique Prime
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Ziyaretçi defteri
 

Farey Sequence

The Farey sequence F_n for any positive integer n is the set of irreducible rational numbers a/b with 0<=a<=b<=n and (a,b)=1 arranged in increasing order. The first few are

F_1 = {0/1,1/1}
(1)
F_2 = {0/1,1/2,1/1}
(2)
F_3 = {0/1,1/3,1/2,2/3,1/1}
(3)
F_4 = {0/1,1/4,1/3,1/2,2/3,3/4,1/1}
(4)
F_5 = {0/1,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,1/1}
(5)

(Sloane's A006842 and A006843). Except for F_1, each F_n has an odd number of terms and the middle term is always 1/2.

Let p/q, p^'/q^', and p^('')/q^('') be three successive terms in a Farey series. Then

 qp^'-pq^'=1
(6)
 (p^')/(q^')=(p+p^(''))/(q+q^('')).
(7)

These two statements are actually equivalent (Hardy and Wright 1979, p. 24). For a method of computing a successive sequence from an existing one of n terms, insert the mediant fraction (a+b)/(c+d) between terms a/c and b/d when c+d<=n (Hardy and Wright 1979, pp. 25-26; Conway and Guy 1996; Apostol 1997). Given 0<=a/b<c/d<=1 with bc-ad=1, let h/k be the mediant of a/b and c/d. Then a/b<h/k<c/d, and these fractions satisfy the unimodular relations

 bh-ak=1
(8)
 ck-dh=1
(9)

(Apostol 1997, p. 99).

The number of terms N(n) in the Farey sequence for the integer n is

N(n) = 1+sum_(k=1)^(n)phi(k)
(10)
= 1+Phi(n),
(11)

where phi(k) is the totient function and Phi(n) is the summatory function of phi(k), giving 2, 3, 5, 7, 11, 13, 19, ... (Sloane's A005728). The asymptotic limit for the function N(n) is

 N(n)∼(3n^2)/(pi^2)=0.3039635509n^2
(12)

(Vardi 1991, p. 155).

Ford circles provide a method of visualizing the Farey sequence. The Farey sequence F_n defines a subtree of the Stern-Brocot tree obtained by pruning unwanted branches (Graham et al. 1994).

The Season 2 episode "Bettor or Worse" (2006) of the television crime drama NUMB3RS features Farey sequences.


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