 Solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

Since

N = 3 p

, its nontrivial difference-of-squares representation is

3 p = (\frac{p + 3}{2})^{2} - (\frac{p - 3}{2})^{2} .

p

is prime, every earlier square difference would give another nontrivial factorization, which is impossible. Thus the first successful value is

r = (p + 3) /2

, and

k = \frac{p + 3}{2} - ⌊ 3 p ⌋ .

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