Arguments about the authority of the papacy eventually return to the question of the primacy of the Bishop of Rome and of the See of Rome. Did early Christians recognize this primacy, or was it something that developed only centuries after our Lord's time and in response to political concerns? These quotations may help you decide.
Clement of Rome
"The Church of God which sojourns in Rome to the Church of God which sojourns in Corinth, to those who are called and sanctified by the will of God through our Lord Jesus Christ. Grace and peace from almighty God be multiplied unto you through Jesus Christ. Owing to the sudden and repeated calamities and misfortunes which have befallen us, we must acknowledge that we have been somewhat tardy in turning our attention to the matters in dispute among you, beloved" (Epistle to the Corinthians, circa A.D. 80]).
"Accept our counsel and you will have nothing to regret. . . . If anyone disobeys the things which have been said by him [Jesus] through us, let them know that they will involve themselves in no small danger. We, however, shall be innocent of this sin and will pray with entreaty and supplication that the Creator of all may keep unharmed the number of his elect." (ibid. 58:2, 59:1).
"You will afford us joy and gladness if, being obedient to the things which we have written through the Holy Spirit, you will root out the wicked passion of jealousy, in accord with the plea for peace and concord which we have made in this letter" (ibid. 63:2).
Ignatius of Antioch
"You [the See of Rome] have envied no one, but others have you taught. I desire only that what you have enjoined in your instructions may remain in force" (Epistle to the Romans 3:1 [A.D. 110]).
"Therefore shall you write two little books and send one to Clement [Bishop of Rome] and one to Grapte. Clement shall then send it to the cities abroad, because that is his duty, and Grapte shall instruct the widows and the orphans. But you shall read it in this city along with the presbyters who are in charge of the Church" (Vision 2:4:3 [circa A.D. 140]).
"For from the beginning it has been your custom to do good to all the brethren in various ways and to send contributions to all the churches in every city. . . Thus custom your blessed Bishop Soter has not only preserved, but is augmenting, by furnishing an abundance of supplies to the saints and by urging with consoling words, as a loving father his children, the brethren who are journeying" (Epistle to Soter [Bishop of Rome] 4:23:9 [inter AD. 166-174]).
"Today we have observed the Lord's holy day, in which we have read your letter [in church]. Whenever we do read it, we shall be able to profit thereby, as also we do when we read the earlier letter written to us by Clement" (ibid. 4:23:11).
"The blessed apostles [Peter and Paul], having founded and built up the church [of Rome], they handed over the office of the episcopate to Linus. Paul makes mention of this Linus in the epistle to Timothy [2 Tim. 4:21]. To him succeeded Anencletus, and after him, in the third place from the apostles, Clement was chosen for the episcopate. He had seen the blessed apostles and was acquainted with them. It might be said that he still heard the echoes of the preaching of the apostles and had their traditions before his eyes. And not only he, for there were many still remaining who had been instructed by the apostles. In the time of Clement, no small dissension having arisen among the brethren in Corinth, the Church in Rome sent a very strong letter to the Corinthians, exhorting them to peace and renewing their faith. . . To this Clement, Evaristus succeeded. . . and now, in the twelfth place after the apostles, the lot of the episcopate [of Rome] has fallen to Eleutherus. In this order, and by the teaching of the apostles handed down in the Church, the preaching of the truth has come down to us" (Against Heresies 3:3:3 [inter AD. 180-190]).
"But since it would be too long to enumerate in such a volume as this the succession of all the churches, we shall confound all those who, in whatever manner, whether through self-satisfaction or vainglory, or through blindness and wicked opinion, assemble other than where it is proper, by pointing out here the successions of the bishops qf the greatest and most ancient church known to all, founded and organized at Rome by the two most glorious apostles, Peter and Paul, that church which has the tradition and the faith which comes down to us after having been announced to men by the apostles. With that church, because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition" (ibid. 3:3:2).
"With a false bishop appointed for themselves by heretics, they dare even to set sail and carry letters from schismatics and b.asphemers to the Chair of Peter and to the principal church [at Rome], in which sacerdotal unity has its source" (Epistle to Cornelius [Bishop of Rome] 59:14 [A.D. 252]).
"In the city of Rome the episcopal chair was given first to Peter, the chair in which Peter sat, the same who was head - that is why he is also called Cephas - of all the apostles, the one chair in which unity is maintained by all. Neither do the apostles proceed individually on their own, and anyone who would [presume to] set up another chair in opposition to that single chair would, by that very fact, be a schismatic and a sinner. . . .Recall, then, the origins of your chair, those of you who wish to claim for yourselves the title of holy Church" (The Schism of the Donatists 2:2 [circa A.D. 367]).
"If the very order of episcopal succession is to be considered, how much more surely, truly, and safely do we number them [the bishops of Rome] from Peter himself, to whom, as to one representing the whole Church, the Lord said, 'Upon this rock I will build my Church, and the gates of hell shall not conquer it.' Peter was succeeded by Linus, Linus by Clement . . . In this order of succession a Donatist bishop is not to be found" (Epistle to Generosus 53:1:2 [A.D. 400]).
"[On this matter of the Pelagians] two councils have already been sent to the Apostolic See [the Bishop of Rome], and from there rescripts too have come. The matter is at an end; would that the error too might be at an end!" (Sermons 131:10 [inter A.D. 391-430]).
"If cases of greater importance are to be heard, they are, as the synod decrees and as happy custom requires, after episcopal judgment, to be referred to the Apostolic See" (Epistle to Victricius[Bishop of Rouen] 2:3:6 [A.D. 404]).
"We exhort you in every respect, honorable brother, to heed obediently what has been written by the most blessed Pope of the city of Rome, for blessed Peter, who lives and presides in his own see, provides the truth of the faith to those who seek it. For we, by reason of our pursuit of peace and faith, cannot try these cases on the faith without the consent of the Bishop of the city of Rome" (Epistle to Eutyches 25:2 [A.D. 449]).