Topology of Numbers · Problem 5

Quadratic Forms — Allen Hatcher

Mathematics — Number Theory · topograph-quadratic-forms

Problem

Find the quadratic form

Q(x,y)

for which

Q(3,5) = Q(4,7) = Q(7,12) = 1

by first drawing a strip in the Farey diagram containing the triangles

\langle ^1/_0, ^0/_1, ^1/_1 \rangle

and

\langle ^3/_5, ^4/_7, ^7/_{12} \rangle

(this can be done using the continued fraction for

^7/_{12}

), then adding the edges of the dual tree that meet these triangles, then filling in values of the topograph starting with the given values.