Matt 1 vs 6 says Solomon was in Christ's genealogy, whereas Luke 3 vs 31 says Nathan was. Nathan and Solomon were both sons of Bathsheba and David (1 Chron 3 vs 4-9).
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The genealogy of Jesus recorded by Matthew traces Jesus' ancestry through the male royal line of David, beginning with his son Solomon, to show that Joseph was in that royal line, which would have been how the Jews (the primary audience to whom Matthew directed his gospel) would have reckoned the fulfillment of prophecies in the Old Testament speaking of the Messiah as a royal descendant of David. (Matthew's focus on his Jewish audience is also shown by the fact that he begins his genealogy with Abraham -- the father of the Jewish nation.) The two problems with that geneaology were that Joseph (although entrusted with Jesus' care as Mary's husband) was not Jesus' biological father, and also that the royal line descended from David had ceased with the curse placed by God on David's descendant Jeconiah (also known as Coniah) (mentioned in Matthew 1:11-12), as recorded in Jeremiah 22:30, because of Jeconiah's wickedness (which had been one of the factors contributing to the tribe of Judah -- the tribe to which David and his descendants belonged -- being carried away into a seventy-year exile in Babylon long before Jesus' birth). The genealogy recorded by Luke in Luke 3:23-38 is the same as Matthew's from Abraham to David, but then indicates Jesus' descent through David's son Nathan (rather than Solomon). In addition, it does not say that Heli begat Joseph -- which would have been impossible (since, as noted in Matthew, Joseph's biological father was named Jacob) -- but that Joseph was the son of Heli. Since it would have been necessary for the fulfillment of Messianic prophecy for Jesus to have been biologically (as well as royally) descended from David, Luke's wording indicates that Heli must have been the father of Mary, and thus the father-in-law of Joseph, which would have made Joseph Heli's son from a legal standpoint, as recorded by Luke. (Luke, who wrote his gospel primarily to a Greek audience that would not have been familiar with Jewish prophecy, also traced Jesus' genealogy all the way back to Adam to show His significance and relationship with respect to all of humanity -- not just the Jews -- from the very beginning of creation.) In this way, Jesus was descended from David (as God had promised David in 2 Samuel 7:16) through both of His lineages, and the curse placed on Jeconiah was avoided.
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