Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 1G / a

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- Solution a

Solution

Write

f = [a, b, c]

. Two integral binary quadratic forms

f

and

g

are equivalent if there is a matrix

U = (α γ β δ) \in SL_{2} (Z)

such that

g (x, y) = f (αx + β y, γ x + δy) .

Equivalently, the group

SL_{2} (Z)

acts on forms by integral unimodular changes of variables.

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