What the Hebrew says. In the chain of fathers running from the Flood to Abraham, Arphaxad's son is Shelah: "And Arpachshad lived five and thirty years, and begat Shelah:" (Genesis 11:12, ASV).
What the Greek says. Brenton's Septuagint puts a different man there: "And Arphaxad lived a hundred and thirty-five years, and begot Cainan." (Genesis 11:12, Brenton). The next verse then gives Cainan his own generation, which the Hebrew has nowhere: "And Cainan lived a hundred and thirty years and begot Sala; and Cainan lived after he had begotten Sala, three hundred and thirty years, and begot sons and daughters, and died." (Genesis 11:13, Brenton).
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And the New Testament has the extra name. Luke's genealogy of Jesus runs back through the same men, and it includes Cainan: "the son of Cainan, the son of Arphaxad, the son of Shem, the son of Noah, the son of Lamech," (Luke 3:36, BSB). The reading is identical in every hosted English version, the King James, the American Standard, the Berean Standard, the World English Bible, the Free Bible Version and the Literal Standard Version all name Cainan there.
So a reader can check the whole thing in three texts, in English, without a word of Greek: Genesis 11 in an ordinary Bible has no Cainan, Genesis 11 in Brenton's Septuagint does, and Luke 3:36 in that same ordinary Bible does. ⚠️ What this does not settle is which reading of Genesis 11 is original, or why Luke has the name. Those are the arguments; this is the evidence they argue about. The 1915 International Standard Bible Encyclopedia notes the insertion as a known feature of the Greek text, describing the Septuagint as "inserting a new name into the genealogical list of Ge 11" (ISBE article: Chronology of the Old Testament).
Sources: Genesis 11:12 (ASV); Genesis 11:12-13 (Brenton); Luke 3:36 (BSB, and identical in KJV, ASV, WEB, LSV, FBV); ISBE 1915, article Chronology of the Old Testament.
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