If we are to think like the first-century hearers, we must recognize that the importance of the number fourteen is that it is a multiple of (that ever-so-important number) seven. Matthew is implying that the entire flow of God's history of creating a people for himself shows that Jesus the Christ is our Sabbath rest.

Forty-two, not Fourteen

Three sets of fourteen is six sets of seven. The operative number here is not really fourteen but forty-two. And no, forty-two is not the answer to life, the universe, and everything; actually it is the precursor to the one who is the meaning of everything. Jesus Christ is the seventh week of history; he is the completion and the culmination; he inaugurates the last days.

In the Lord Jesus, God's redemptive work comes to a rest. This does not mean that God becomes inert, but that that the finality and totality are invested in him, and history never moves beyond him, as he lives on in his indestructible life, ruling from on high.

Old Testament Prophecies

I am not aware of prophecies that reference the number fourteen. However, when this is seen as a matter of the seventh seven, all the significance of seven and its square come into play. There are a handful of references to forty-two in the Old Testament, but none of them shed a lot of light on this situation. The most relevant prophecy may be Daniel's reference to the week of weeks:

Now listen and understand! Seven sets of seven plus sixty-two sets of seven will pass from the time the command is given to rebuild Jerusalem until a ruler—the Anointed One—comes. Jerusalem will be rebuilt with streets and strong defenses, despite the perilous times. —Daniel 9:25 (NLT)

Admittedly, Daniel is giving a different perspective on the narrative than Matthew, but the same theological meaning of the seventh seven: the Messiah will be the fulfillment of all things.

New Testament Prophecies

The meaning of forty-two is further clarified by the last book of prophecy as the time of the dominion of the nations over God's people:

But do not measure the outer courtyard, for it has been turned over to the nations. They will trample the holy city for 42 months. —Revelation 11:2 (NLT)

And:

Then the beast was allowed to speak great blasphemies against God. And he was given authority to do whatever he wanted for forty-two months. —Revelation 13:5 (NLT)

The Messiah is the one who brings this time to an end, defeating the enemies of God and rescuing his people. Upon his arrival, the time of trampling is over.

Conclusion

In Jon Ericson's answer, he links to an article by Bob Deffinbaugh, who points out the strong connections between Matthew 1 and Genesis 1. Indeed, what we see here is that just as God created progressively over six days, so God revealed himself progressively for six weeks, until on the seventh God himself appeared on earth and became our rest.

There have been several proposed reconciliations of the Matthew and Luke genealogies. Among the popular ones are:

• Matthew's genealogy traces legal heirs; Luke's traces biological ancestors.
• Matthew's genealogy traces the ancestry of Joseph; Luke's traces the ancestry of Mary. This view takes the phrase "as was supposed of Joseph" in 3:23 as a parenthetical expression.
• Matthew's genealogy traces the ancestry of Mary; Luke's traces the ancestry of Joseph. This view assumes the phrase "Joseph the husband of Mary" was originally written as "Joseph the father of Mary". This view is much less common than the previous two; no existing ancient biblical texts read "Joseph the father of Mary".

The reason Luke has more generations than Matthew is because Matthew has left some out in order to split the generations into three sets of 14. We can see evidence of this in the middle section, tracing Solomon to Jechoniah:

Matthew 1:7-11

…and Solomon the father of Rehoboam, and Rehoboam the father of Abijah, and Abijah the father of Asaph, and Asaph the father of Jehoshaphat, and Jehoshaphat the father of Joram, and Joram the father of Uzziah, and Uzziah the father of Jotham, and Jotham the father of Ahaz, and Ahaz the father of Hezekiah, and Hezekiah the father of Manasseh, and Manasseh the father of Amos, and Amos the father of Josiah, and Josiah the father of Jechoniah and his brothers, at the time of the deportation to Babylon.

Compare this to the same genealogy in 1 Chronicles;

1 Chronicles 3:10-16 [emphasis mine]

The descendants of Solomon: Rehoboam, Abijah his son, Asa his son, Jehoshaphat his son, Joram his son, Ahaziah his son, Joash his son, Amaziah his son, Azariah his son, Jotham his son, Ahaz his son, Hezekiah his son, Manasseh his son, Amon his son, Josiah his son. The sons of Josiah: Johanan the firstborn, the second Jehoiakim, the third Zedekiah, the fourth Shallum. The descendants of Jehoiakim: Jeconiah his son, Zedekiah his son;

Asaph in Matthew is Asa in 1 Chronicles and Uzziah in Matthew is Azariah in 1 Chronicles. However, Joram's great-grandson Azariah from Chronicles is listed as his son in Matthew. This technique for shortening lengthy genealogies is known as telescoping.