Mathematics of the Jewish Calendar/The recurrence period of the calendar
The recurrence period of a calendar is the period after which it will always repeat exactly, i.e. two days of the same date that many years apart will always be on the same day of the week. This means that the interval must be an exact number of weeks, or a multiple of seven days.
For the Julian calendar, the recurrence period is 28 years. This is also true for the Gregorian calendar unless the period covers the boundary between February and March in a century year which is not a leap year, such as 1900 or 2100. Even allowing for such years, the recurrence period is 400 years.
The recurrence period for the Jewish calendar is vastly longer: 689,472 years, or 36,288 cycles of 19 years. Any days of the same date this many years apart will always be 251,827,457 days or exactly 35,975,351 weeks apart, so any two dates separated by this number of years must fall on the same day of the week.
Proof of the recurrence period
The time of the molad after a cycle of 19 years or 235 months exceeds a full week by 2 days, 16 hours, 595 chalakim, or 69,715 = 13,943 x 5 chalakim.
In a whole week there are 7 x 24 x 1080 = 36,288 x 5 chalakim.
It follows that the calendar cannot recur until the passage of 36,288 cycles of 19 years or 689,472 Jewish years.
Another way to express this is that the average year over a 19 year cycle has length 35975351/98496 days. Thus after 98,496 years the Molad will be at the same time of day as before, and hence after 98,496 x 7 = 689,472 years the Molad will be at the same time of day on the same day of the week as before.
The 247 year cycle
It has often been claimed that the calendar always repeats itself after 247 years, or 13 cycles of 19 years. This is because after 247 years, the computed Molad of Tishri is short by only 905 chalakim (about 50 minutes) of being at the same time on the same day of the week as before. Such a small difference will rarely make any difference to the year type, so corresponding dates in these years will nearly always be on the same weekday. This period is often called the Cycle of Nachshon Gaon, as it was attributed to Rabbi Nachshon, Gaon of Sura 871-9, by Abraham ibn Ezra (early 12th century).
However, "nearly always" is not the same as always. Periods of 247 Hebrew years are usually 90216 = 12888 x 7 days long, an exact number of weeks, so two dates separated by 247 years are on the same day of the week. However, the period may last for 90215 or 90214 days, not an exact number of weeks, and the calendar does not recur.
The last time that Rosh Hashana was not on the same day of the week as 247 years earlier was 5663 (1902CE), when it was Thursday not Saturday. The next time this happens will be 5848 (2087CE), when it will be Saturday not Monday.
Some printed copies of the Jewish law code known as the Arba Turim ("Four rows") give lists of the weekday of Rosh Hashana assuming the correctness of the 247 year cycle. The errors were pointed out by Rabbi Hezekiah di Silo (17th century CE) in his book P'ri Chadash ("New fruit").
There are even closer (but still not exact) correspondences. For example:
- After 190 cycles or 190x19 = 3610 years there is an increase of 730 chalakim.
- After 190+13 = 203 cycles or 203x19 = 3857 years there is a decrease of 175 chalakim.
- After 1002 = 203x5-13 cycles or 1002x19 = 19038 years there is an increase of 30 chalakim.
- After 5213 = 1002x5+203 cycles or 5213x19 = 99047 years there is a decrease of 25 chalakim.
- After 1002+5213 = 6215 cycles or 6215x19 = 118085 years there is an increase of 5 chalakim; this is the best possible approximate recurrence, since the number of chalakim in a complete cycle is a multiple of 5 so a difference of less than 5 chalakim between starts of cycles is impossible.
The Jewish-Gregorian calendar correspondence cycle
Thus the Jewish calendar repeats after 689,472 Jewish years, while the Gregorian calendar repeats after 400 Gregorian years.
400 Gregorian years are 146,097 days or 20,871 weeks. It follows that the correspondence cycle between the Hebrew and Gregorian calendars, the time interval after which any given Jewish date is guaranteed to fall on the same Gregorian date as before, is 20,871 of these 689,472 year cycles or 14,389,970,112 Jewish years. These amount to 5,255,890,855,047 days or 14,390,140,400 Gregorian years. This is approximately the current age of the universe according to the Big Bang model.