# 5.4. What years are leap years? (Persian Calendar)

Since the Persian year is defined by the astronomical vernal equinox,
the answer is simply: Leap years are years in which there are 366 days
between two Persian new year's days.

However, basing the Persian calendar purely on an astronomical
observation of the vernal equinox is rejected by many, and a few
mathematical rules for determining the length of the year have been
suggested.

The most popular (and complex) of these is probably the following:

The calendar is divided into periods of 2820 years. These periods are
then divided into 88 cycles whose lengths follow this pattern:

29, 33, 33, 33, 29, 33, 33, 33, 29, 33, 33, 33, ...

This gives 2816 years. The total of 2820 years is achieved by
extending the last cycle by 4 years (for a total of 37 years).

If you number the years within each cycle starting with 0, then leap
years are the years that are divisible by 4, except that the year 0 is
not a leap year.

So within, say, a 29 year cycle, this is the leap year pattern:

```Year            Year              Year              Year
0  Ordinary      8  Leap          16  Leap          24  Leap
1  Ordinary      9  Ordinary      17  Ordinary      25  Ordinary
2  Ordinary     10  Ordinary      18  Ordinary      26  Ordinary
3  Ordinary     11  Ordinary      19  Ordinary      27  Ordinary
4  Leap         12  Leap          20  Leap          28  Leap
5  Ordinary     13  Ordinary      21  Ordinary
6  Ordinary     14  Ordinary      22  Ordinary
7  Ordinary     15  Ordinary      23  Ordinary
```

This gives a total of 683 leap years every 2820 years, which
corresponds to an average year length of 365 683/2820 = 365.24220
days. This is a better approximation to the tropical year than the
365.2425 days of the Gregorian calendar.

The current 2820 year period started in the year AP 475 (AD 1096).

This "mathematical" calendar currently coincides closely with the
purely astronomical calendar. In the years between AP 1244 and 1531
(AD 1865 and 2152) a discrepancy of one day is seen twice, namely in
AP 1404 and 1437 (starting at vernal equinox of AD 2025 and 2058).
However, outside this period, discrepancies are more frequent.

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