The Farey series FN is the set of all fractions in lowest terms between 0 and 1 whose denominators do not exceed N, arranged in order of magnitude. For example, F6 is
N is known as the order of the series. Farey was a British geologist who in 1816 published the statement to the effect that in the Farey series the middle of any three successive terms is the mediant of the other two. The proof has been eventually supplied by Cauchy. The series nonetheless bears the name of Farey.
To see why the statement is correct, start with F1 which is
When N is prime, the rule adds N-1 fractions. In general, φ(N) fractions are added. For all reducible fractions m/N will have appeared in one of the earlier series. Check this with F7
The Farey series furnishes another proof of an important corollary of Euclid's algorithm: for integers m and n with
An absolutely marvelous geometric interpretation of the Farey series has been invented by Lester R. Ford.
- A. Beck, M.N. Bleicher, D. W. Crowe, Excursions into Mathematics, A K Peters, 2000
- J.H.Conway and R.K.Guy, The Book of Numbers, Springer-Verlag, NY, 1996.
- R.Graham, D.Knuth, O.Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, 1994.
- Modular Arithmetic
- Chinese Remainder Theorem
- Euclid's Algorithm
- Euclid's Game
- Binary Euclid's Algorithm>
- gcd and the Fundamental Theorem of Arithmetic
- Extension of Euclid's Algorithm
- Stern-Brocot Tree
- Farey series
- Pick's Theorem
- Fermat's Little Theorem
- Wilson's Theorem
- Euler's Function
- Divisibility Criteria
- Equivalence relations
- A real life story
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