The first blow comes from examining the First Ecumenical Council, held in 325AD at Nicaea. In Canons 6 and 7 it reads:
Canon 6 The ancient customs of Egypt, Libya and Pentapolis shall be maintained, according to which the bishop of Alexandria has authority over all these places since a similar custom exists with reference to the bishop of Rome. Similarly in Antioch and the other provinces the prerogatives of the churches are to be preserved.These two canons are very revealing on this matter: Canon 6 mentions Alexandria, Antioch, and Rome. Canon 7 mentions Aelia, which was the name the Romans gave to Jerusalem after they destroyed it in 70AD, but notice that Jerusalem doesn't even have the status of a Metropolitan. (In this case a Metropolitan is what the EO would classify as a Patriarch.) Also notice, no mention of Constantinople here. That's because it didn't exist yet.
Canon 7 Since there prevails a custom and ancient tradition to the effect that the bishop of Aelia [Jerusalem] is to be honoured, let him be granted everything consequent upon this honour, saving the dignity proper to the metropolitan.
So Constantinople and Jerusalem weren't even Patriarchates, and thus there was not a Pentarchy at the time of Nicaea, meaning such a concept is not Apostolic nor part of the pre-Nicene Church. (Some people mistakenly read Canon 6 as saying Rome is on the same level as Alexandria and Antioch, but This Article shows that's an incorrect reading of the text. Catholics deny that Rome ever was a Patriarchate since the Bishop of Rome is above the status of a Patriarch.)
The second blow comes from the Patriarchal schism at the Fourth Ecumenical Council, held in 451 at Chalcedon. Here the Patriarchs of Alexandria and Antioch refused to accept the decisions of the Council, considering the Chalcedonian idea of "two natures" to be heretical. As a result, they split off, considering themselves the true Church and defending orthodoxy. They are known as the Oriental Orthodox - not to be confused with the Eastern Orthodox. They remain out of communion with both Rome and Eastern Orthodox to this day. Again, this entails that the theory of a Pentarchy is false, since the only window of time such a Pentarchy could have existed was between Nicaea and Chalcedon, a span of just around 100 years long.
What many people don't realize, and I myself didn't realize for a while, is that shortly after the Oriental Orthodox schism, the Patriarch of Constantinople appointed his own Greek-speaking heritage Patriarchs of Alexandria and of Antioch. So while some might say there has always been an Eastern Orthodox Patriarch of Alexandria and of Antioch, this is very misleading since these were invented after Chalcedon. The Oriental Orthodox are generally Arabic-speaking heritage Churches and constitute the original Patriarchs of Alexandria (known popularly as the Coptic Church) and Antioch (known popularly as Syriac Orthodox).
With these two details in mind, it becomes clear that the Pentarchy is a historical fiction. The idea was invented around the time of Chalcedon (451AD) as a way for Constantinople to slowly take over the Church and push Rome aside. The 4-agaist-1 idea arose when Constantinople used it's political power to take control of Alexandria, Antioch, and Jerusalem and use this phony 'majority' to make Rome look unreasonable and out of place.