No. The gospels of Luke, John and Matthew, and the Old Testament prophesies, as well as some other secular historical texts all confirm that Jesus was born in Bethlehem, a few miles outside the city of Jerusalem in Israel/Palestine.
The idea that Jesus was born 'at' Migdal Eder as if Migdal Eder was a different place, are wrong. The phrase Migdal Eder means "tower of the flock" . In ancient times this was a military tower to view into the valley on the edge of Bethlehem to protect the city.
Many such towers are recorded in the Old Testament with the Bethlehem tower first mentioned in Genesis 35:21. The fact that Jacob is recorded as having moved his flock from the Tower to Bethlehem (Ephratah) means that the tower was very close to present day Bethlehem. As Bethlehem was believed to have been biger in Jesus' day, this establishes that Migdal Edar, "the tower of the flock" was actually within Bethlehem in Bible times.
No, Migdal Eder was a small town outside of Jerusalem that was used to specify a distance when discussing various types of offerings. Please see the related link.
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