The Genealogies in Matthew and LukeAs was mentioned earlier, Matthew started his gospel by giving the genealogy of Jesus (Matthew 1:1-17) from Jesus to Abraham. Luke also gave a genealogy of Jesus tracing it all the way to Adam. (Luke 3:23-38). Both these genealogies trace Jesus' ancestry from Joseph's side. There are many problems with these genealogies:
So according to Matthew, Joseph's father was Jacob but according to Luke he was Heli! The discrepancies doesn't stop here. The table below summarizes in tabular form the generations from Jesus to David as given by both gospels. A quick glance at the table will reveal the glaring inconsistencies in both list. With the exception of Shealtiel and Zerubbabel [a], none of the other names coincide! And worse, Luke has forty three generations from David to Jesus, Matthew lists only twenty eight, giving a discrepancy of fifteen generations. Assuming an average of 25 to 30 years per generation, this brings the disparity of about four centuries between the genealogies.  In short, to put it bluntly, the two genealogies contradict one another at almost every turn.
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Attempt No.1: The LevirateThe first explanation is that the two lists does give the genealogy from Joseph's side but one list the legal heritage while the other the natural. This, they say, is due to the Jewish religious custom of the Levirate (Deuteronomy 25:5-10). The levirate was the law that decreed that if a man died without leaving any offspring, it is the duty of the deceased man's brother to impregnate his brother's wife to give him offsprings to perpetuate the family line. Thus the ancients say, one line of the genealogy gives the actual legal father, the other gives the natural father (from the levirate). The difficulty in this explanation is obvious, we are to suppose that the levirate affected all the generations except two (Schealtiel and Zerubbabel) from David to Jesus. As Guignebert concludes, such an explanation is clearly "absurd." 
Attempt No.2: Africanus' "Half Brothers" Plus LevirateAnother explanation comes from the third century Christian writer Julius Africanus (c160-c240). According to Africanus, Jacob (Joseph's father as given in Matthew) and Heli (Joseph's father as given in Luke) were brothers. When Heli died childless, Jacob impregnated his sister-in-law, and presto!, both Heli and Jacob are Joseph's father! Heli being the legal father while Jacob is the biological father via the levirate. The obvious question then, is why do these two brothers have different fathers: Heli's father is Matthat and Jacob's father is Matthan. Africanus' solution here is typical apologetic nonsense: he claimed that Jacob and Heli were half brothers! They shared the same mother who after the death of her first husband, Matthan, remarried this time to Matthat! The explanation is rather strange and sounds unreal. Did Africanus supplied any proof of this? No, but evidence to the early Christians are of no consequence. As Africanus himself says:
Thus was how the inconsistency was reconciled in ancient times. With convoluted explanations based on hypothetical levirate and second marriages.
This explanation was eventually abandoned by the Christian apologists.
Attempt No.3: Mary's GenealogyAround the end of fifteenth century Annius of Viterbo suggested another alternative explanation to this discrepancy. This "explanation" maintains that the genealogy in Matthew applied to Joseph while the one in Luke applied to Mary! This explanation does not hold water. In the first place both gospels state explicitly that they are tracing Jesus' ancestors from Joseph's side (Matthew 1:16; Luke 3:23-see above). In fact Luke, the one the apologists claim traces the ancestry from Mary, always refers to Joseph whenever he talks about ancestry of David:
Furthermore Luke, by making Mary the relative of Elizabeth (Luke 1:36), whom he gave as a descendent of priestly family of Aaron [b] (Luke 1:5) immediately makes Mary a member of that family.
In the second place the Jews do not admit to transmission of birthright by the mother, as St. Jerome rightly said, "It is not the custom of the scriptures to count women in their genealogies." Thus a genealogy traced from Mary's side is of no value in determining the descendents from David.  The fact that this discredited medieval explanation is still the same one used by some believers todays only serves to underline the bankrupt state of Christian theology.
The contradictions remain. At best, only one of these genealogies can be true, at worse, both could be false. Can we find out one way or the other? Yes we can!
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From David to the "exile" ends with Jeconiah. The numerical significance perhaps lies in Matthew trying to show that every fourteen generations something spectacular happens: for fourteen generations after Abraham was David and fourteen generations after that was the exile and so fourteen generations after the exile the messiah was born. All this sounds impressive. However Matthew has been less than honest in making the genealogy fit his numerological scheme. In the second list of fourteen generations (from David to Jeconiah) we have the seemingly innocent verse:
Matthew had skipped three generations from Joram to Uziah to keep his nicely balanced numerology. For we know from the Old Testament that Joram was actually the great great grandfather of Uzziah:
Matthew has left out three generations (Ahaziah, Joash and Amaziah) to make the genealogy conform to his numerology. The cavalier method in which Matthew uses some of his sources which are known to us does not inspire confidence in the veracity of his entire list.  If Matthew can simply delete names that do not conform to his preconceived opinion, what guarantee do we have that he did not add names where he could not find any?
There is another mistake in Matthew's list:
Jeconiah is just another from of the name Jehoiachin and we know from the Old Testament that Josiah was Jeconiah's grandfather:
Again we see Matthew skipping a generation to make his numerological scheme work.  Actually even as it stands in Matthew there is actually only thirteen generations from the exile to Jesus (for Jehoiachin was already counted in the second fourteen generation [See the table above].). The best that can be stated regarding the number of generations are that there are fourteen from Abraham to David, eighteen from David to the exile and thirteen from the exile to Jesus.  Neither a pretty nor satisfying numerological relationship!
The conclusion regarding Matthew's handling of the genealogy is most aptly stated by Guignebert:
While the early genealogies had their source in the Old Testament, the generations after the exile from Abiud to Joseph is taken from a source no longer known to us. Bearing in mind the way Matthew used available material it is not impossible that some of these names could well be fictitious.
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Thus according to Luke, Shelah was the grandson of Arphaxad. However, the Hebrew Old Testament explicitly stated that Arphaxad was the father of Shelah.
Only in the Greek Septuagint do we find the name Cainan inserted between Arphaxad and Shelah. There is however another name that did not appear in any extant text known to us, i.e. neither in the Hebrew Bible nor in the Greek Septuagint. That name appears in Luke 3:27, Rhesa
Rhesa, in the above passage is the father of Joanan and the son of Zerubbabel. This makes Joanan the grandson of Zerubbabel. Now Joanan is just another form of the name Hananiah and we know from the Old Testament that he was the son of Zerubbabel.
There is no such person as Rhesa and Luke had simply inserted another generation to the list.  It remains an open question whether Luke's mistakes in inserting the names of Rhesa and Cainan were accidental or purposeful. It should be noted that the number of generations from God to Jesus in Luke's list is 77 and we know that the messianic number is seven. So perhaps Luke, like Matthew, dabbled in numerology. 
In Luke the generations given after Nathan, with the exception of Zerubbabel, Shealtiel and Joanan consist of unknown names. We do not know where Luke get these names from. It should also be mentioned that even with known names we have discrepancies between the two list. In Matthew the father of Shealtiel was given as Jeconiah [Jehoaichin], in Luke the father is given as Neri. (One gets a feeling the levirate explanation may well be revived by fundamentalists to explain this.)
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