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Friends,you must keep in mind If Peter had said he was writing from Rome, then no doubt, the Romans would have begun an intensive search for him.
Christians were under constant persecution by both the Jews and the Romans from the very beginning and had to practice the faith underground in the homes of believers, and in the catacombs of Rome.

More writings(bear with me because my electricity keeps going on and off due to the 50 plus mph winds in upstate NY today)

"Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its pre- eminent authority, that is, the faithful everywhere, inasmuch as the apostolical tradition has been preserved continuously by those [faithful men] who exist everywhere." Irenaeus, Against Heresies, 3:3:2 (A.D. 180).

"And he says to him again after the resurrection, 'Feed my sheep.' It is on him that he builds the Church, and to him that he entrusts the sheep to feed. And although he assigns a like power to all the apostles, yet he founded a single Chair, thus establishing by his own authority the source and hallmark of the (Church's) oneness. No doubt the others were all that Peter was, but a primacy is given to Peter, and it is (thus) made clear that there is but one flock which is to be fed by all the apostles in common accord. If a man does not hold fast to this oneness of Peter, does he imagine that he still holds the faith? If he deserts the Chair of Peter upon whom the Church was built, has he still confidence that he is in the Church? This unity firmly should we hold and maintain, especially we bishops, presiding in the Church, in order that we may approve the episcopate itself to be the one and undivided." Cyprian, The Unity of the Church, 4-5 (A.D. 251-256).

"Thereupon Victor, who presided over the church at Rome, immediately attempted to cut off from the common unity the parishes of all Asia, with the churches that agreed with them, as heterodox; and he wrote letters and declared all the brethren there wholly excommunicate..." Pope Victor I [regn. A.D. 189-198], in Eusebius EH, 24:9 (A.D. 192).

"Stephen, that he who so boasts of the place of his episcopate, and contends that he holds the succession from Peter, on whom the foundations of the Church were laid...Stephen, who announces that he holds by succession the throne of Peter." Pope Stephen I [regn. A.D. 254-257], Firmilian to Cyprian, Epistle 74/75:17 (A.D. 256).

"I beseech you, readily bear with me: what I write is for the common good. For what we have received from the blessed Apostle Peter s, that I signify to you; and I should not have written this, as deeming that these things were manifest unto all men, had not these proceedings so disturbed us." Pope Julius [regn. A.D. 337-352], To the Eusebians, fragment in Athanasius' Against the Arians, 2:35 (c. A.D. 345).