Ptolemaic and Roman
According to Roman writer Censorinus, the Egyptian New Year's Day fell on July 20 in the Julian Calendar in 139 CE, which was a heliacal rising of Sirius in Egypt. From this it is possible to calculate that the previous occasion on which this occurred was 1322 BC, and the one before that was 2782 BCE. This latter date has been postulated as the time when the calendar was invented, but Djer's reign preceded that date.
In 238 BCE, the Ptolemaic rulers decreed that every 4th year should be 366 days long rather than 365. The Egyptians, most of whom were farmers, did not accept the reform, as it was the agricultural seasons that made up their year. The reform eventually went into effect with the introduction of the "Alexandrian calendar" by Augustus in 26/25 BCE, which included a 6th epagomenal day for the first time in 22 BCE. This almost stopped the movement of the first day of the year, 1 Thoth, relative to the seasons, leaving it on 29 August in the Julian calendar except in the year before a Julian leap year, when a 6th epagomenal day occurred on 29 August, shifting 1 Thoth to 30 August.
Read more about this topic: Egyptian Calendar
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