Blue Eyes On An Island

An island has 1,000,000 inhabitants. Each inhabitant has blue eyes or brown eyes. There are no reflective surfaces on the island, so no inhabitant knows their own eye color. The custom of the island also prevents anyone from talking or communicating about eye colors. The inhabitants are otherwise completely logical, able to deduce everything that can be deduced in an hour or so. If anyone knows their eye color, they must leave the island at the next midnight, never to return. All inhabitants know this paragraph.

An outsider that wants to wreck havoc arrives at the island and declares that there are a composite number of inhabitants who have blue eyes, before promptly fleeing. What is the minimum possible number of midnights after which the island gets completely deserted, no matter how many inhabitants with blue eyes there were initially?

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