By the fourth century that changed and they had become very interested. By that time also they were very much interested in the Virgin Mary whose status was greatly elevated within the church. There would never have been a Christmas without an increase of importance in the person and the worship of Mary. The church faced with the competition from the winter religious solstice festivals like Saturnalia, and other religions like Mithraism decided not to celebrate Christmas on its correct historical date but on competitive reasons against the other religions of the time. The Christians were not interested in accuracy after all that is the realm of science and not religion.
Religions are more interested in the conversion of souls.
The majority of the Roman subjects were accustomed to celebrating at that time of the year some god whether it is called by its Mesopotamian name of Marduk or its Greek Zeus, Roman Saturn or the Persian Mithras. The chief rival of Christianity at that time was however Mithraism. Mithraism had been enjoying widespread popularity within the Roman empire. Conversion to Christianity was slow and difficult. The Church fathers despised the pagan Gods and the winter festivals but were also faced with the difficult task of conversion.