The apparent contradiction is that Shelah (or Salah) have two (biological) father: Arphaxad and Cainan.
Several Old Testament passages clearly present Arphaxad as Shelah’s father: Genesis 10:24, 11:12, 1 Chronicles 1:18.
Genesis 11:12
When Arphaxad had lived 35 years, he became the father of Shelah
In contrast, Luke 3:35–36 records that Arphaxad is the father of Cainan, who then is the father of Shelah.
Luke 3:35-36
…the son of Shelah, the son of Cainan, the son of Arphaxad…
In other words, the key difference is whether Arphaxad is the father or grandfather of Shelah.
This apparent discrepancy can be addressed in two main ways:
- “Father” as a broader ancestral term
In biblical genealogies, “father” and “son” often refer to ancestral relationships, not necessarily direct parentage. For example, in Genesis 32:9, Jacob refers to Abraham as his “father,” even though Abraham was his grandfather. It is possible that Cainan was a real ancestor and the genealogy is simply telescoped in Genesis, omitting his name. - Cainan as a scribal insertion error
Another possibility is that Luke’s inclusion of Cainan results from a rare scribal error. A phrase like “son of Cainan” may have been inadvertently copied from later in the genealogy (Luke 3:37) into verse 36. While the Septuagint (Greek Old Testament) does include Cainan in Genesis and 1 Chronicles, the earliest known Hebrew manuscripts and early Septuagint copies before around AD 220 do not mention him at all (see https://creation.com/cainan-can-you-explain-the-difference-between-luke-336-and-genesis-1112)