 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/11h/c/solution

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

Let

P = p_{1} p_{2} \dots p_{k + 1}

be the product of the first

k + 1

primes, and take

n = P^{a}

a

τ (P^{a}) = (a + 1)^{k + 1}, (lo g P^{a})^{k} = a^{k} (lo g P)^{k} .

Their ratio is

\frac{( a + 1 ) ^{k + 1}}{a ^{k} ( l o g P ) ^{k}} = \frac{a ( 1 + 1/ a ) ^{k + 1}}{( l o g P ) ^{k}} ⟶ \infty.

Thus every sufficiently large $a$ gives $τ (P^{a}) \geq (lo g P^{a})^{k}$ , producing infinitely many such $n$ .

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