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Agur was a compiler (not the actual Sacred Author) of part of the collection of Proverbs in the book of Proverbs, which forms part of the Old Testament. His contribution to this book is mainly in Chapter 30 of the book. His name could suggest that he is free from earthly transgressions. It could also mean 'He who understood the will of the Lord'.
Lemuel is only mentioned in Chapter 31 of the book of Proverbs, as a King of Israel, most probably referring to King Solomon, son of King David. Still, there is no clear evidence of this. The meaning of his name in Hebrew suggests the he was the one who "belongs to God". This could imply that he was a Spiritual leader, and not an actual king of the Jews, but this remains pure speculation. Some modern studies have also linked Lemuel to the King of Assyria.