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Most, if not all, Orthodox Christians believe in Peter’s supremacy. Many Ecumenical Councils have affirmed this. They simply believe that it was a personal authority and did not pass on to the bishops of Rome. This is refuted by the traditions of Rome dating back to the early Church. Tradition states that when Peter died, the Church of Rome gathered around St. Paul to seek direction. Paul, knowing of the Apostle’s casting lots to have St. Matthias replace Judas after his suicide, commanded the Church to cast lots to determine who would replace Peter in his Apostolic office.
The lots were cast and the lot fell on Linus, the elder of the two bishops which Paul, himself, had installed to lead the Church of Rome when he had first visited the City. After Paul died, the Church of Rome continued the tradition and cast lots to choose who would replace Paul in his Apostolic (or ministerial, depending upon who you ask) office. The lot fell to Clement, the younger of the two bishops which Paul had installed in the Church of Rome years earlier. Through the tradition of lots, Cletus followed Linus when he was martyred. After Cletus’ eventual death, the lots were cast again and Clement was chosen. While Clement was the 3rd (4th if you count Peter), it was from his papacy onward that the Papacy claimed the Apostolic office of both Peter and Paul.
The authority of the bishops of Rome as successors of Peter’s Apostolic Office was, indeed, recognized in the Early Church. St. Irenaeus of Lyons (c.130 - c.202) used Peter’s authority to reject the Gnostic belief of a secret knowledge of salvation. If that secret knowledge existed, then the successors of Peter, in his office as leader of the Apostles, would know of that secret knowledge. They did not know the existence of any such secret. Thus, the Gnostic beliefs were not true.
Hope this helps!
God Bless,
Ben