" It is certain that Saint Paul had preached at Rome, and he was probably put to death there about 67 during the reign of Nero. It seems likely, as well, that Saint Peter had visited Rome and had also been martyred there. About Peter's actual position at Rome, however, and about the position of the early Roman bishops, the historical record is silent. What is unquestioned is that by the 3d century the Roman bishops were representing themselves as having succeeded to the primacy that Peter had enjoyed among the apostles and as wielding within the universal church a primacy of authority in doctrinal matters."LINK
There is no clear historic record of Peter as bishop of Rome..
Church Fathers Cyprian, Origen, Cyril, Hilary, Jerome, Ambrose, Augustine,did not apply the "thou are Peter" verse to mean Peter was the pope . .
In reading the book of Acts, it is clear that the only person upon whom any ‘primacy’ was conferred by the apostles was James, the eldest half brother of Jesus.