To me, this Lilius/Clavius algorithm is the most beautiful and elegant,
because it gives a clear understanding of how the moon behaves on the Gregorian calendar
and how it determines the Easter date.
I consider all other algorithms as just very smart arithmetic tricks to find the same result
(except for the Gauss method, which is an incomplete algorithm since it still needs a table).
Explanation:
is the golden number, a sequence number in the 19year Metonic cycle or moon cycle.
Meton, an ancient Greek, had discovered that in 19 tropical years there are nearly exactly
235 synodic months (= full moons).
is just the number of the century.
takes account for the difference between the Gregorian and the Julian calendar
(the Gregorian calendar has 3 leap years less every 4 centuries).
is a correction for the inaccuracy of the Metonic cycle, because that is not exactly
19 years. It shifts 1 day in 310 (Julian) years, which is nearly the same as 8 days
in 25 (Gregorian) centuries.
takes account for the weekday of 21 March. Since a year is 1 day longer than 52 weeks,
it comes 1 weekday later every year plus a leap day every 4 years, resulting in 5 days
every 4 years (do you recognize that your own birthday does just the same thing?).
C is used to correct E for the Gregorian nonleap centuries.
is the epacta, the age of the moon at the start of the year. Since a tropical year
is 11 days longer than 12 synodic moons (a so called moon year) the full moon comes 11 days
sooner each year,
which is clearly visible in the algorithm. In the same time the corrections by the D and C
components are applied. The second (correcting) step in the calculation of F results from
the fact that a synodic moon is not exactly 30 days but a bit less (29.53059), and the 11
days difference between the just mentioned moon year and the tropical year is also not
exact (10.89).
is either the first full moon since the vernal equinox on 21 March or the last one before.
If it is before then a synodic month of 30 days is added.
Finally, the resulting Sunday following this Paschall Full Moon is calculated very easily.
In this Lilius/Clavius algorithm, one can see some periodic cycles:
The already mentioned Metonic or moon cycle of 19 years.
The C component of the algorithm corrects the epacta for the Gregorian nonleap
centuries with 3 days per 400 years. That will cycle through F if it becomes a
multiple of 30 (the (approximated) synodic month) in an integer number of centuries,
i.e. in 4000 years.

The weekday of 21 March depends on the main cycle of the Gregorian calendar itself,
which is 400 years (that's exactly an integer number (20871) of weeks).
Thus, the Gregorian Easter algorithm has a total cycle duration equal to the
Smallest Common Multiple of 19, 4000, 37500, and 400,
which is 5 700 000 years.
