Mathematics of the Jewish Calendar/The 19 year cycle
In the Jewish calendar, it is usual to give the year as A.M. (Anno Mundi, Year of the World). This is calculated from the data given in the Tanach (Old Testament); the calculation is given in a book called the Seder Olam (Order of the World). The six days of creation described in Genesis chapter 1 have to be in some year, so they are assumed to be the last six days of Year 1. The first Shabbat (Saturday), described in Genesis chapter 2, was also the first day (Rosh Hashanah) of year 2AM.
An earlier tradition was that the first five days of creation were the last five days of Year 1 and the Friday was the first day of year 2AM. However, this is not in accordance with the present calendar rules, which forbid the year to start on Monday, Wednesday or Friday.
According to the Jewish calculation, creation was in the autumn of 3761 BCE, slightly later than the traditional Christian figure of 4004 BCE. Thus the year 5701 began in 1940 in the Gregorian calendar and ended in 1941. Using that as a starting point, it is easy to convert years between the two calendars, remembering always that the new years do not coincide so a year in one calendar will always overlap with two years in the other.
A year is a leap year if the remainder (in AM) on division by 19 is 3, 6, 8, 11, 14, 17 or 0. Thus 5703, 5706, 5708, 5711, 5714, 5717 and 5719 were leap years with 13 months each, but not the years between, which only had 12 months. This rule is based on the wikipedia:Metonic cycle, which assumes that 19 years exactly equal 235 lunar months.
The 19 year cycle is quite accurate. Assuming 365.24219 days for the tropical year, to do better with a fixed cycle would require a cycle of 182 years containing 2,251 months. However, it is not perfect; see The long-term accuracy of the calendar.
Length of a 19 year cycle
A 19-year cycle may have 6939, 6940 or 6941 days. It is theoretically possible for it to have 6942 days, but this is very rare, as will be discussed later. However, 19 consecutive years, some in one cycle and some in the next, may have only 6938 days, for example 5719-5737 and 5790-5808, and 19 years of total length 6942 days are more common than exact cycles of this length.
Two days on the same date but exactly 19 years apart may thus be 6938, 6939, 6940, 6941 or 6942 days apart. In the Julian or Gregorian calendars, two days on the same date but exactly 19 years apart may be 6939 or 6940 days apart; they may be only 6938 days apart in the Gregorian calendar if the period includes say 1900 or 2100, which are not leap years. Thus, your 19th birthday by the two calendars is not necessarily on the same day, but may be one or two days different, and in rare cases the Jewish date may be three or possibly four days after the civil one.