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How long in advance can dates of moon phases be accurately predicted?

When researching an article on Chinese New Year I found a few websites that predict the dates of Chinese New Year (CNY) tens - or sometimes even hundreds - of years in advance. I´m wondering how many ...
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When do natural leap years occur, on years based on the northern winter solstice?

I'm designing a new calendar and I want the first day of the year to be the first day after the longest night of year in the northern hemisphere. Under this plan, "natural" leap years, where the first ...
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Tropical year is 27s/year off from Gregorian year?

http://hpiers.obspm.fr/eop-pc/models/constants.html states the tropical year is $365.242190402$ days. The Gregorian calendar's average year is $365.2425$ days (every 4th year a leap year, except ...