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The Bible is not clear, and these questions have never been answered definitively. the best we can do is repeat the theories of those who have tried to answer this before us.
Most people assume there were three, based on the fact that there are three gifts, but this isn't conclusive.
The Huffington Post put out an article that examines this question. It can be found online here.
It mentions several theories and includes the mention of a piece of Apocrypha called Revelation of the Magi, which purports to be a the wise men's personal testimony. Whether it's truly what it claims to be is just as inconclusive as any other answer you're likely to get on this question.
Upvote:9
In Persia is the city called Saveh, from which the three Magi set out when they came to worship Jesus Christ. Here, too, they lie buried in three sepulchres of great size and beauty. Above each sepulchre is a square building with a domed roof of very fine workmanship. The one is just beside the other. Their bodies are still whole, and they have hair and beards. One was named Beltasar, the second Gaspar, and the third Melchior. Messer Marco asked several of the inhabitants who these Magi were; but no one could tell him anything except that they were three kings who were buried there in days gone by. ... Let me tell you finally that one of the three Magi was from Saveh, one from Hawah, and the third from Kashan.
Marco Polo The Travels: The Travels of Marco Polo, Translated and with an Introduction By Ronald Latham, Penguin Books Ltd., Harmondsworth, England, UK, 1976. pp. 58-60.
It is believed they were from Hecatompylos (now Damghan: DaiMogan: convent of Mogi) the capital city of Parthians. But Marco Polo cites three different cities. It is correct since Mogi in Parthians time were part of the ruling class and used to convene in a Mogistan at the capital from far and near cities. Mogistan had the role of advisory parliament for Parthian kings and had a mixture of aristocracy and priests. So three Mogi could be priests or princes or barons.