Why do Matthew and Luke give different genealogies of Jesus?

The passages

And Jacob begat Joseph the husband of Mary, of whom was born Jesus, who is called Christ.

Matthew 1:16

And Jesus himself began to be about thirty years of age, being (as was supposed) the son of Joseph, which was the son of Heli.

Luke 3:23

So all the generations from Abraham to David are fourteen generations; and from David until the carrying away into Babylon are fourteen generations; and from the carrying away into Babylon unto Christ are fourteen generations.

Matthew 1:17

The problem

Matthew gives Joseph's father as Jacob. Luke gives him as Heli. From David down the two lists share almost no names: Matthew goes through Solomon and the kings of Judah, Luke through Nathan, another son of David. Matthew counts fourteen generations three times, but his own list has gaps; he skips several kings that appear in 1 and 2 Chronicles. Luke has 77 names and goes back to Adam.

What believers and scholars say

Luke gives Mary's line. Since the second century some have read Luke's "son of Heli" as describing Mary's father, with Joseph named as the legal heir. The Greek of Luke 3:23 does not say this, and a genealogy that names only men and passes over Mary would be an odd way to record hers. Still, the idea is old and widely held.

Levirate marriage. Julius Africanus, writing around AD 220, proposed that Jacob and Heli were half-brothers, and that one died childless so that the other raised a son in his name. It is possible, though Africanus said he was relaying a family tradition he could not prove.

Each genealogy has a purpose. Johnson and Brown argue that ancient genealogies served theology more than record-keeping. Matthew presents Jesus as son of David and son of Abraham, with five women named, four of them with irregular histories. Luke emphasizes Jesus as son of Adam, for all humanity. On this view, harmonizing the names misreads what the lists were for.

Where this leaves it

Both lists are probably doing a theological job, so a reader who treats them as legal archives is asking them for something they were not built to give. The harmonies are old and not impossible, but the text never signals them. Neither genealogy is the sort of document a modern historian could use to check Jesus's ancestry.