Hatcher-Topology-of-Numbers · Problem 4.1.5

Quadratic Forms

Mathematics — Algebraic Topology · topograph

Problem

Find the quadratic form Q(x,y)Q(x,y) for which Q(3,5)=Q(4,7)=Q(7,12)=1Q(3,5) = Q(4,7) = Q(7,12) = 1 by first drawing a strip in the Farey diagram containing the triangles ⟨1/0,0/1,1/1⟩\langle {}^1/_0, {}^0/_1, {}^1/_1 \rangle and ⟨3/5,4/7,7/12⟩\langle {}^3/_5, {}^4/_7, {}^7/_{12} \rangle (this can be done using the continued fraction for 7/12{}^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.

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