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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/1g/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 1G / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Every proper equivalence class of a positive definite form contains a unique reduced form

[a, b, c]

, where

∣ b ∣ \leq a \leq c,

with

b \geq 0

when equality occurs in either inequality. Its discriminant is

b^{2} - 4 a c = - 80.

For a reduced positive definite form,

3 a^{2} \leq 80

, so

1 \leq a \leq 5

. Also

b

is even and

a c = 20 + \frac{b ^{2}}{4} .

Checking the finitely many possibilities with

∣ b ∣ \leq a \leq c

gives

[1, 0, 20], [3, 2, 7], [3, - 2, 7], [4, 0, 5] .

Each coefficient triple has greatest common divisor one. By uniqueness of reduced representatives, these are four distinct classes. Hence the requested number is

4 .

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