Email Gateway (May 18 2025 at 13:31): Email Gateway (May 18 2025 at 13:31):From: Tobias Nipkow <nipkow@in.tum.de>
Farey Sequences and Ford Circles
Lawrence C. Paulson
The sequence F_n of Farey fractions of order n has the form ... where the
fractions appear in numerical order and have denominators at most n. The
transformation from F_n to F_{n+1} can be effected by combining adjacent
elements of the sequence F_n, using an operation called the mediant. Adjacent
(reduced) fractions a/b < c/d satisfy the unimodular relation and their mediant
is (a+c)/(b+d). A Ford circle is specified by a rational number, and interesting
consequences follow in the case of Ford circles obtained from some Farey
sequence F_n. The formalised material is drawn from Apostol's Modular Functions
and Dirichlet Series in Number Theory.
https://www.isa-afp.org/entries/Farey_Sequences.html
Enjoy!
Last updated: May 30 2025 at 04:27 UTC