The text is first mentioned, critically, in the True Account of the anti-Christian writer Celsus (c. 178 AD), and therefore would have been contemporary with the surviving, and much more famous, dialogue between the convert from paganism Justin Martyr and Trypho the Jew.
[1] he has chosen one that is worthless, which although it could be of some help to the simple-minded multitude in respect of their faith, certainly could not impress the more intelligent, saying: "I know a work of this sort a Controversy between one Papiscus and Jason..." ...
The third source is a letter (mistakenly included in the works of Cyprian) to a certain "Bishop Vigilius" (not Vigilius of Thapsus(de)) describing a translation from Greek to Latin by an otherwise unknown Celsus, (given the sobriquet Celsus Africanus by scholars), which also describes the Dialogue, including the information that Jason himself was a convert from Judaism, and the ending - that Papiscus is convinced and asks for baptism.
[4][5] A recent discovery in St. Catherine's monastery at Mount Sinai provides more text quoted from the Dialogue.
[7] Maximus the Confessor (7th century, or possibly 6th century if mainly the work of John of Scythopolis as suggested by Hans Urs von Balthasar[8]), notes that Clement of Alexandria, in the (now lost) sixth book of his Hypotyposes ascribes the Dialogue to Luke the Evangelist, though Maximus himself ascribes the authorship to Ariston of Pella, in Latin Aristo Pellaeus, an author whom Eusebius mentions in connection with emperor Hadrian and Simon bar Kokhba.