Open Matthew 1 and read the genealogy. Then open Luke 3 and read verse 23 onward.
They do not match.
They run through different sons of David. They give Joseph a different father. Matthew counts down from Abraham to Christ; Luke counts back from Christ all the way to Adam. The middle sections are not the same list of men.
If you have never noticed this, notice it now, because somebody will eventually show it to you as though it were a bomb.
This is not new
First, calm down. The Church has known about this for as long as there has been a Church.
Julius Africanus was writing about the two genealogies in the third century, and his work is attached to the relevant passages in the Church Fathers section of this site. The Fathers did not discover the problem in embarrassment and hide it. They wrote about it openly and proposed explanations.
So the framing that modern scholarship uncovered a contradiction the Church was concealing is simply false as a matter of history. It was discussed in the open, early.
The explanations
Several accounts have been offered and they are not mutually exclusive.
Levirate marriage. The Law required that if a man died childless, his brother take the widow and raise up offspring in the dead man's name. That produces a man who has a legal father and a biological father who are different people. Run two genealogies through such a family, one tracing legal descent and one tracing physical descent, and you get two different names in the same slot with neither being wrong. This is essentially the line Africanus takes.
One line is Mary's. On this reading Matthew gives Joseph's line and Luke gives Mary's, with Joseph named because genealogies were reckoned through men. Notice that Luke says Jesus was, as it was supposed, the son of Joseph. That parenthesis is doing something.
Legal versus natural throughout. Matthew traces the royal succession, the line of legal right to the throne of David, which does not always follow biological descent. Luke traces actual physical descent.
I am not going to tell you which of these is certainly correct. I will tell you that any one of them dissolves the contradiction, and that the objection depends on assuming an ancient genealogy must work like a modern birth certificate.
What each one is actually arguing
This is the part I find more interesting than the harmonisation, and almost nobody talks about it.
Matthew starts at Abraham and comes down through David and the kings. He is writing to people who know that Scripture, and he is making a claim: this man is the son of David and the son of Abraham. He is the King. He is the promise to Israel arriving.
Luke goes the other direction and does not stop at Abraham. He keeps going, back past the covenant, past the flood, to Adam, and then one step further, to God.
Read that ending and see what he has done. Luke is writing for the whole world, and he has just tied Christ not merely to Israel but to the human race as such.
That is recapitulation, which I wrote about in the article on Irenaeus. Christ is the second Adam. Luke's genealogy is that doctrine in the form of a list.
So the two genealogies are not two failed attempts at the same document. They are two arguments. Matthew's is about the throne. Luke's is about the race. Both are true and neither is redundant.
Trace it yourself
Reading a genealogy as a wall of names is miserable. Seeing it as a structure is not.
- Open the biblical family tree and find David, then follow the lines forward.
- Read Matthew 1 and Luke 3 side by side and mark where they diverge and where they meet.
- Read what the Fathers say on both passages in the Church Fathers section, Julius Africanus included.
- Notice the four women in Matthew's list and go read their stories. He put them there on purpose and it is not because they were respectable.
That last one is worth a whole evening. Matthew is making a point about who this King came through, and it is not a flattering list of ancestors. It is a list of exactly the sort of people He came to save.