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Luke's historical details for the birth of Jesus are problematic.
Luke 1:5, "In the days of King Herod," who died in 4 BC.
Luke 2:1, "In those days a decree went out from Emperor (or Caesar) Augustus that all the world should be registered," places the birth anytime between 27 BC to 14 AD, the length of the very long reign of Augustus.
Luke 2:2, "This was the first registration and was taken while Quirinius was governor of Syria," places the date at 6 or 7 AD.
Most scholars agree that Luke and Matthew's reference to "In the days of King Herod," is the most reliable reference and that Jesus was born sometime before Herod's death in 4 BC.
Note that Matthew 1:18-25 and 2:1-12, make no mention of Nazareth or a census.
Note that in Luke there is no mention of how Joseph and Mary traveled from Nazareth to Bethlehem. The distance between these two villages is approximately 70 miles or 130 kilometers; a journey of at least 7 to 10 days by foot. It is likely that neither place was a "city" or even a "town," as each would have about 100 adult residents.