Today is February 29, 2024, a leap day. It occurs once during leap years once every 4 years when the year is divisible by 4, with some exceptions. Every 400 years, 3 of these years are not leap years. In particular, every century year (e.g. 1800, 1900, ...) that is not divisible by 400 is not a leap year. So there are 97 leap years every 400 years.
The reason for that is that the mean tropical year which describes a complete cycle of seasons is the basis of the Gregorian calendar. Currently (as the mean tropical year drifts over thousands of years), the mean tropical year is about 365 days, 5 hours, 48 minutes, 45 seconds = 365.2421875 days. Therefore 97/400 = 0.2425 approximates well the fraction 0.2421875 thus the current rule of leap years. However, there is a small mismatch which might needs to be compensated over hundreds of years.
Since 2421875/10000000 = 31/128, it might be better to have 31 leap years every 128 years, but the resulting rule might be too complicated for years written in decimal. However, if we represent years in binary (which how it is represented in modern computers anyway), then a possible rule is relatively simple. We could define a leap year as a year where in binary, the least significant 2 bits is 0, and the least significant 7 bits are not all 0. Under this scheme, years divisible by 4 are leap years, except for a year that is divisible by 128. For instance, 1920 and 2048 will not be leap years in this scheme.
Looking at the continued fraction convergents (which describe the best rational approximations) of 2421875/10000000: 1/4, 7/29, 8/33, 31/128, there are other rules possible, but besides 1/4 (once every 4 years) none of the other intermediate convergents lead to an interesting rule.
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