Farey sequence

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, 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. 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 } 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. The history of 'Farey series' is very curious — Hardy & Wright (1979) 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) 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.

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Related concepts (5)

Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named.

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Farey sequence