A rigorous proof in Shakespearean dialogue

The prompt

Write a proof of the fact that there are infinitely many primes; do it in the style of a Shakespeare play through a dialogue between two parties arguing over the proof.

Expected result

A mathematically valid proof of the infinitude of primes (typically Euclid's argument), delivered as period-flavored dialogue between two characters — correct on both the math and the style.

Why it works

Combines a rigorous constraint (the proof must actually be correct) with a stylistic one (Shakespearean dialogue) — tests whether a model sacrifices correctness for style or holds both at once.