 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/12h/d/solution

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

For each

x

, let

h (x)

be an effectively obtained index of the partial function

y ⟼ {0, undefined, y < x^{2}, y \geq x^{2} .

The existence of the total computable index transformation

h

follows from the S-m-n theorem. By the recursion theorem, some

m

satisfies

f_{m, 1} ≃ f_{h (m), 1}

. Therefore its domain is

W_{m} = {0, 1, \dots, m^{2} - 1},

and hence

∣ W_{m} ∣ = m^{2} .

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