000RM.dvi

1.3 The Farey sequence 107

Exercise

1.Can a lattice triangle be equilateral? Why?

2.Can a lattice polygon be regular? Why? [You may make use of the

nontrivial fact thatthe only values ofnfor whichsinπnis rational

isn=6.]

3.ForB =3, 4, 6, 8, 9, give an example of a lattice triangle with

exactly one interior point andBboundary points.^2

4.Give an example of an equilateral lattice hexagon.

5.How many terms does the Farey sequenceFnhave? [Hint: Give

the answer in terms of the Eulerφ-function].

6.The Farey polygonPnis the lattice polygon whose vertices, taken

in order, are the origin and the points(k, h)forhkin the Farey se-

quenceFn. Here isP 6.

Find the area ofP 6 and that ofPnfor a generaln.

(^2) In [Weaver], it is shown that these are the only possible values ofBifI=1.