Leap Week Cycles a Multiple of 100 Years

Dear Calendar People

I’m aware that the Gregorian 400-year cycle is the shortest accurate calendar cycle that is both a multiple of 100 years and a whole number of weeks so can be used as a leap week calendar cycle. It is the only such cycle less than 1000 years.

The 900-year cycle of the Modified Julian Calendar proposed for the Eastern Orthodox church is not a whole number of weeks and so has to be multiplied by 7 to get a leap week cycle. Irv mentioned the 6300-year cycle.

Counted in centuries and leap weeks, these two cycles are mixer cycles:

4 centuries 71 leap weeks

63 centuries 1118 leap weeks

63*71 – 4*1118 = 1.

So we can use these to build other cycles without fear of missing out any shorter cycles whose mean year lies in between..

Mix (-11,1)  19 centuries 337 leap weeks – mean year  365.241579 days

Mix (-10,1)  23 centuries 408 leap weeks – mean year 365.241739 days

Mix (-9,1) 27 centuries 479 leap weeks – mean year 365.241852 days

Mix (-8,1) 31 centuries 550 leap weeks – mean year 365.241936 days – Fifty 62-year cycles

Mix (-7,1) 35 centuries 621 leap weeks – mean year 365.242 days exactly

Mix (-6,1) 39 centuries 692 leap weeks – mean year 365.242051 days

Mix (-5,1) 43 centuries 763 leap weeks – mean year 365.242093 days

...

Mix (-2,1) 55 centuries 976 leap weeks – mean year 365.242182 days

Mix(-1,1) 59 centuries 1047 leap weeks – mean year 365.242203 days

Mix (0,1) 63 centuries 1118 leap weeks – mean year 365.242222 days

...

Mix (3,1) 75 centuries  1331 leap weeks – mean year 365.242267 days

...

Mix (8,1) 95 centuries 1686 leap weeks – mean year 365.242316 days

Mix (9,1) 99 centuries 1757 leap weeks – mean year 365.242323 days

...

Mix (42,1) 231 centuries 4100 leap weeks – mean year 365.242424 days

I note that one 95 century cycle is one week longer than five 19-century cycles.

Karl

12(07(23