The Genealogy of Jesus

Differences between Matthew and Luke

The genealogy of Jesus Christ in Matthew seems to contradict the genealogy in Luke. In Matthew, it says that Jacob begot Joseph, the husband of Mary (Mat 1:16 (NKJV)). But in Luke it says that Joseph was the son of Heli (Luk 3:23). 

This is explained by Eusebius of Caesarea who lived in the fourth century: 
Joseph's mother's first husband Heli died childless, after which Heli's half-brother Jacob married the widow according to the law of levirate marriage (Deu (5 Mos) 25:5–6, Luk 20:28). Thus Jacob became the physical father of Joseph (Mat 1:16), while Joseph was counted as the son of Heli (Luk 3:23) according to the law. 

Heli and Jacob were half-brothers because they had different fathers and grandfathers (Luk 3:24, Mat 1:15). This is explained by having the same mother. She was first married to either Heli's father or Jacob's father and she had a son
(who later fathered Heli or Jacob). Then her first husband died and she remarried with the other one of Heli's father/Jacob's father, and she had another son
(who later became the other father to Jacob or Heli). It was allowed for a widow to remarry with a man who isn't a relative of her first husband if she had a son with her first husband. This means that both genealogies in Matthew and Luke are of Joseph, none are of Mary. Jesus' legal father Joseph was of the lineage of David
(Luke 1:27b, 2:4), see The Virgin Birth.

This explanation is in detail described in the video: The Genealogy of Jesus Christ.

Ungodly and evil kings are mentioned in Jesus' genealogy, such as Manasseh and Amon (Mat 1:10). That there are sinners in Jesus'genealogy points to the fact that Jesus came into the world to save sinners (1 Tim 1:15) and is dependent on God's grace and mercy, see Grace and Mercy in God's Attributes. 

See also The Curse of Jeconiah.

Jesus had to be sinless in order to pay for our sins through his physical death on the cross. This means that he also had to be without sin when he was born as a man. Adam acted as a representative of mankind. Eve didn't. All mankind fell into sin because of Adam (Rom 5:12, 19, Psalm 51:5 (7 in some translations)). Adam brought death into the world because of his disobedience, see 
The Biblical Creation and the Fall of Man. Sin entered the human race, Adam's failure became our failure. This sin is inherited through the father and not through the mother. Therefore, the virgin birth is required so that Jesus can be born as a man without sin (Mat 1:22-23, Isaiah 7:14). If Jesus hadn't been sinless, his sacrifice on the cross wouldn't have been accepted by God the Father because of his holiness, see God is holy in God's Attributes.

Genealogies in the Bible

• Mat 1:1–17
• Luke 3:23–38
• Gen (1 Mos) 5:1–32, 11:10–26
• Ruth 4:18–22
• 1 Chron 1:1–4, 17–27, 34, 2:1–15

Cainan:

Gen (1 Mos) 11:12 doesn't mention Cainan: 
"When Arpachshad had lived 35 years, he fathered Shelah." 
Luke 3:35b–36a mentions Cainan:
"the son of Shelah, the son of Cainan, the son of Arphaxad". 
The Old Testament is based on the Masoretic Text which doesn't include Cainan. However, Cainan is included in the Septuagint, but not in older texts of the 
Septuagint. Cainan is also not included in Luke's oldest texts. 
The well-known historian Josephus doesn't include Cainan, nor does 1 Chron 1:18.

Conclusion: Cainan shouldn't be included in Luke 3:36a.

Hezron's offspring:

Luke 3:33a mentions Admin and Arni instead of Ram:
"the son of Amminadab, the son of Admin, the son of Arni, the son of Hezron".
Mat 1:3b–4a: "Hezron the father of Ram, and Ram the father of Amminadab".
Ruth 4:19: "Hezron fathered Ram, Ram fathered Amminadab".
1 Chron 2:9-10a: "The sons of Hezron that were born to him:
Jerahmeel, Ram, and Chelubai. Ram fathered Amminadab". 

The English translations of Luke 3:33a in NASB, CSB, and NIV use Ram instead of Arni. CSB and NIV omit Admin entirely. Admin (Adokimos) in NASB means not approved or worthless. Probably it should be Ram instead of Arni.
Admin may have been left out due to embarrassment to the family.

See also Some stories in the gospels at the end of The Four Gospels.