Praise be to Allaah.
We thank you for this question and we appreciate your interest in the prayer times in Islam. In answer to your question, those who live close to the Arctic Circle, where they have continual night or continual day for several months, should look to the closest city to them where night and day are distinct in each twenty-four hour period, then they should do the five daily prayers based on the timing in that city.
Your interest in the matter and your keenness to ask questions and find out gives us the hope that you will embrace this religion and practise its rituals. We wish you well and hope for guidance for you.