The Genealogy of Jesus

I have followed Jesus for over forty years, and read the Bible regularly. In all that time, I have found passages I didn't understand but never one that I found to be in contradiction to another. Now the other day, I was reading in Luke 3 about when John the Baptist starts his ministry, and it begins with a firm historical peg: "in the fifteenth year of the reign of Tiberius Caesar, Pontius Pilate being governor of Judea, and Herod being tetrarch of Galilee...". I like those touch-points with history. Then Luke moves on rather quickly to where Jesus is baptised, and John sees the Spirit descend on him like a dove. Then Luke goes into another sort of historical peg, stating that Jesus "began to be about thirty years old" - was it his birthday? - "and being (as was supposed) the son of Joseph, who was the son of Heli," etc. All the way back to Adam, the son of God. Now, I think it's cool that his complete genealogy is given, 75 generations from Adam. It sounds so little these days, and it's the kind of thing that leads people to calculate the beginning of the world at 4:00pm (Daylight Savings Time?) on a Sunday or something. Anyway, it is only around 30 generations since he was born to the present, so a lot can happen in this world in that kind of time. But wait, here's my problem: in Matthew, the gospel begins with a genealogy going the other way, only starting with Abraham. Matthew goes through David and on to Joseph, "the husband of Mary, of whom was born Jesus". The problem is that in Matthew, we go from David to Solomon and onward, whereas in Luke we go from David to his son Nathan, and onward. People sometimes are called by different names, but that is not the case here. Nathan and Solomon are two distinct sons of King David. So what is going on here?