Optimum Intercalation of YLM Calendars
by Karl Palmen

To say that a calendar has optimum intercalation is equivalent to saying that the number of long years in any n-year period is n*L/Y either rounded up or round down to a whole number.

Proposition: All YLM calendars have optimal intercalation

Proof: Consider an n-year period with l long years. The quantity [y.L/Y] (from rule 2) is l greater for the last year of the period than for the last year before the start of the period. If the rounding is ignored the difference becomes n.L/Y regardless of which year the period begins. So the actual difference is given by

l = [(y+n).L/Y] - [y.L/Y]

where y is the last year before the start of the period.

Consider the remainders (y+n).L mod Y and y.L mod Y. Their difference is at most Y-1. If r is the first of these remainders minus the second then (noting that r/Y is what one needs to subtract to get rid of the []s)

l = n.L/Y - r/Y

Since l is a whole number and r/Y is strictly between -1 and 1, l must be n.L/Y either rounded up or rounded down. Q.E.D.

The sign of r determines whether the rounding is up or down. If the remainder goes up over the period, the rounding is down and if the remainder goes down, the rounding is up. Also r/Y is the estimate a of how much the new year runs ahead against the seasons. So the remainder y.L mod Y for any year y is an estimate of the earliness of next year's new year.

Last modified: 1999-05-05 CE / 102-25-02-21 MP