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Do we know the identities of the wise men?

No. Storytellers have given them the names of Balthazar, Caspar and Melchoir. Wikipedia gives us the origin of the names:

Traditions identify a variety of different names for the Magi. In the Wester Christian church their names have been commonly known since the 8th century as Balthazar, Caspar and Melchoir. These derive from an early 6th century Greek manuscript in Alexandria. The Latin text Collectanea et Flores continues the tradition of three kings and their names and gives additional details. This text is said to be from the 8th century, of Irish origin.

Painters have given them all different ethnic origins, like English, African and Asian. This cannot be true. The Bible says they all came from the East and the country east of Israel was the Persian/Babylonian area. These were well educated and well read people who would have the best access to ancient Jewish prophecies because they once conquered Israel.

Since the Bible is silent on their identity, it apparently is not important enough to guess.