Short answer
No. As an EU citizen you don't need a visa or residence permit to live in the Azores, but after three months you must register with the local municipality (câmara municipal) to get a CRUE certificate. Bring ID, proof of address and proof of sufficient means or work. Rules can change, so confirm details with the câmara or AIMA before you move.
General information, not legal or tax advice. Rules and amounts change: confirm with official sources.
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