There are two major discrepancies I want to point out between Matthew and Luke in their Nativities which are as follows:
1) Matthew's story has Joseph and Mary living in Bethlehem when Jesus was born while Luke places their home in Nazareth and attempts to place them in the traditional Bethlehem by an unhistorical census (see Luke ii.1-5).
2) Matthew mentions a flight to Egypt to make a scriptural connection with Hosea xi.1 (see Matthew ii.13-15), while Luke completely omits this tale and doesn't allow time in his chronology for a flight to Egypt.
Exposing the Errancy
The starting setting:
Matthew narrates that Jesus is born in Bethlehem and magi arrive to worship him since they 'saw his star' (Matthew ii.1-12). Jesus is anywhere around 2 years old at this time as indicated by Matthew ii.16. There is no mention of Luke's shepherds who visited when Jesus was newborn at the stable (see Luke ii.7-20). When the magi in Matthew arrive in Bethlehem to worship Jesus, Matthew states they came to 'the house' (Matthew ii.11), indicating Mary and Joseph's home was in Bethlehem (they're still there two years after his birth)--much unlike Luke's story where they're there temporarily for a census and Jesus was born on a farm because there wasn't room anywhere else during the bustle and hustle of the census. In fact, Luke directly states that Mary and Joseph were living in Nazareth (Luke i.26-27; ii.4).
Flight to Egypt:
Matthew narrates that after Herod decides to kill Jesus shortly after the Magi's arrival (this entire event is omitted by Luke and Herod is never mentioned at all), Joseph is warned in a dream by an angel to go to Egypt immediately, where the holy family stays until the death of Herod (Matthew ii.13-18). Remember Jesus is anywhere around two years old when they leave for Egypt.
Out of Egypt:
Matthew narrates that after Herod died an angel came to Joseph again and tells him to go back to Israel (Matthew ii.19-21). Joseph had a mind to go back to Judea (presumably back to Bethlehem), but was afraid to do so because Herod's son now reigned there, so instead he goes to Nazareth in Galilee. This is how the account reads:
But when he heard that Archelaus was reigning over Judea in place of his father Herod, he was afraid to go there. Then after being warned by God in a dream, he left for the regions of Galilee,
and came and lived in a city called Nazareth This was to fulfill what was spoken through the prophets: "He shall be called a Nazarene." (Matthew ii.22-23)
These verses further highlight that Joseph's home was in Bethlehem in Judea (or else why would he want to go back there?) and that he came to live in Nazareth in Galilee to fulfill some non-existent prophecy about Jesus being called a Nazarene. Jesus was about two years old when they left for Egypt and upon the return and the subsequent move to Nazareth, he must have been even older. In any case, he and his family made their home in Nazareth when he was a young boy according to Matthew. This directly contradicts Luke who tells us that Mary and Joseph were already living in Nazareth even before Jesus was born (see above) and were only in Bethlehem for a census. After Jesus was born in Bethlehem, named and circumcised the 8th day after his birth (Luke ii.21) Luke writes:
And when the days for their purification according to the law of Moses were completed, they brought Him up to Jerusalem to present Him to the Lord
(as it is written in the Law of the Lord, "EVERY firstborn MALE THAT OPENS THE WOMB SHALL BE CALLED HOLY TO THE LORD"),
and to offer a sacrifice according to what was said in the Law of the Lord, "A PAIR OF TURTLEDOVES OR TWO YOUNG PIGEONS." (Luke ii.22-24)
These laws concerning circumcision and purification can be found in Leviticus which reads:
"Speak to the sons of Israel, saying: 'When a woman gives birth and bears a male child, then she shall be unclean for seven days, as in the days of her menstruation she shall be unclean.
'On the eighth day the flesh of his foreskin shall be circumcised.
'Then she shall remain in the blood of her purification for thirty-three days; she shall not touch any consecrated thing, nor enter the sanctuary until the days of her purification are completed. (Leviticus xii.1-4)
It goes on to prescribe what to bring as a sacrifice to the Temple/sanctuary, as Luke states, but mark the seven days and the thirty-three days. If you add them you get forty days. So after Mary's uncleanliness of forty days, she then enters the Temple (in Jerusalem, which is in Judea if you don't know, close to Bethlehem) and offers her sacrifice, just as Luke states. So Jesus is only forty days old. But mark what Luke states after this process was finished:
When they had performed everything according to the Law of the Lord, they returned to Galilee, to their own city of Nazareth. (Luke ii.39)
Luke says they returned to Nazareth, where they were already staying. There's no mention of a flight to Egypt, and Joseph and Mary return to Nazareth with Jesus who's a baby not much more than a month old--not a young boy as in Matthew's Nativity. They go there because they stay there, not to fulfill any prophecy as in Matthew (who's trying hard to make Jesus fulfill as many 'prophecies' as he can to appeal to the Jews to whom he's obviously writing so that they can believe he is the Messiah and the Son of God).
1) Matthew's story has Joseph and Mary living in Bethlehem when Jesus was born while Luke places their home in Nazareth and attempts to place them in the traditional Bethlehem by an unhistorical census (see Luke ii.1-5).
2) Matthew mentions a flight to Egypt to make a scriptural connection with Hosea xi.1 (see Matthew ii.13-15), while Luke completely omits this tale and doesn't allow time in his chronology for a flight to Egypt.
Exposing the Errancy
The starting setting:
Matthew narrates that Jesus is born in Bethlehem and magi arrive to worship him since they 'saw his star' (Matthew ii.1-12). Jesus is anywhere around 2 years old at this time as indicated by Matthew ii.16. There is no mention of Luke's shepherds who visited when Jesus was newborn at the stable (see Luke ii.7-20). When the magi in Matthew arrive in Bethlehem to worship Jesus, Matthew states they came to 'the house' (Matthew ii.11), indicating Mary and Joseph's home was in Bethlehem (they're still there two years after his birth)--much unlike Luke's story where they're there temporarily for a census and Jesus was born on a farm because there wasn't room anywhere else during the bustle and hustle of the census. In fact, Luke directly states that Mary and Joseph were living in Nazareth (Luke i.26-27; ii.4).
Flight to Egypt:
Matthew narrates that after Herod decides to kill Jesus shortly after the Magi's arrival (this entire event is omitted by Luke and Herod is never mentioned at all), Joseph is warned in a dream by an angel to go to Egypt immediately, where the holy family stays until the death of Herod (Matthew ii.13-18). Remember Jesus is anywhere around two years old when they leave for Egypt.
Out of Egypt:
Matthew narrates that after Herod died an angel came to Joseph again and tells him to go back to Israel (Matthew ii.19-21). Joseph had a mind to go back to Judea (presumably back to Bethlehem), but was afraid to do so because Herod's son now reigned there, so instead he goes to Nazareth in Galilee. This is how the account reads:
But when he heard that Archelaus was reigning over Judea in place of his father Herod, he was afraid to go there. Then after being warned by God in a dream, he left for the regions of Galilee,
and came and lived in a city called Nazareth This was to fulfill what was spoken through the prophets: "He shall be called a Nazarene." (Matthew ii.22-23)
These verses further highlight that Joseph's home was in Bethlehem in Judea (or else why would he want to go back there?) and that he came to live in Nazareth in Galilee to fulfill some non-existent prophecy about Jesus being called a Nazarene. Jesus was about two years old when they left for Egypt and upon the return and the subsequent move to Nazareth, he must have been even older. In any case, he and his family made their home in Nazareth when he was a young boy according to Matthew. This directly contradicts Luke who tells us that Mary and Joseph were already living in Nazareth even before Jesus was born (see above) and were only in Bethlehem for a census. After Jesus was born in Bethlehem, named and circumcised the 8th day after his birth (Luke ii.21) Luke writes:
And when the days for their purification according to the law of Moses were completed, they brought Him up to Jerusalem to present Him to the Lord
(as it is written in the Law of the Lord, "EVERY firstborn MALE THAT OPENS THE WOMB SHALL BE CALLED HOLY TO THE LORD"),
and to offer a sacrifice according to what was said in the Law of the Lord, "A PAIR OF TURTLEDOVES OR TWO YOUNG PIGEONS." (Luke ii.22-24)
These laws concerning circumcision and purification can be found in Leviticus which reads:
"Speak to the sons of Israel, saying: 'When a woman gives birth and bears a male child, then she shall be unclean for seven days, as in the days of her menstruation she shall be unclean.
'On the eighth day the flesh of his foreskin shall be circumcised.
'Then she shall remain in the blood of her purification for thirty-three days; she shall not touch any consecrated thing, nor enter the sanctuary until the days of her purification are completed. (Leviticus xii.1-4)
It goes on to prescribe what to bring as a sacrifice to the Temple/sanctuary, as Luke states, but mark the seven days and the thirty-three days. If you add them you get forty days. So after Mary's uncleanliness of forty days, she then enters the Temple (in Jerusalem, which is in Judea if you don't know, close to Bethlehem) and offers her sacrifice, just as Luke states. So Jesus is only forty days old. But mark what Luke states after this process was finished:
When they had performed everything according to the Law of the Lord, they returned to Galilee, to their own city of Nazareth. (Luke ii.39)
Luke says they returned to Nazareth, where they were already staying. There's no mention of a flight to Egypt, and Joseph and Mary return to Nazareth with Jesus who's a baby not much more than a month old--not a young boy as in Matthew's Nativity. They go there because they stay there, not to fulfill any prophecy as in Matthew (who's trying hard to make Jesus fulfill as many 'prophecies' as he can to appeal to the Jews to whom he's obviously writing so that they can believe he is the Messiah and the Son of God).