“In the fourth year of the reign of Ptolemy and Cleopatra, Dositheus, who said that he was a priest and a Levite, and his son Ptolemy brought to Egypt the preceding Letter about Purim, which they said was authentic. It had been translated by Lysimachus son of Ptolemy, one of the residents of Jerusalem.”
In a curious note, this book or a letter about Purim was brought to Egypt. The time frame is very clear, during the reign of Ptolemy. However, the question is which Ptolemy? The original Ptolemy was a Macedonian guard of Alexander the Great who was the governor of Egypt in 323 BCE. He later declared himself king and thus established a Ptolemaic dynasty that lasted until 47 BCE. There were a number of kings and queens named Ptolemy and Cleopatra. The first mention of them would be Ptolemy V and Cleopatra I from (202-181 BCE). This would put this translation around the year 198 BCE, right in the middle of the Greek Septuagint work from around 250-132 BCE. We do not know anything about Dositheus but he may have been a Jewish Levite priest. There are 3 or 4 famous people with the name of Lysimachus in Egypt. If it was the son of one of the Ptolemy kings, he might have died around 181 BCE, as the brother of Ptolemy V, it would be a good fit for this translation.