To say that a calendar has optimum intercalation is equivalent to saying that the number of long years in any

Optimum Intercalation of YLM Calendars by Karl Palmen

n-year period isn*L/Yeither rounded up or round down to a whole number.

Proposition: All YLM calendars have optimal intercalationProof: Consider an

n-year period withllong years. The quantity[y.L/Y](from rule 2) islgreater 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 becomesn.L/Yregardless of which year the period begins. So the actual difference is given by

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

yis the last year before the start of the period.Consider the remainders

(y+n).L mod Yandy.L mod Y. Their difference is at mostY-1. Ifris the first of these remainders minus the second then (noting thatr/Yis what one needs to subtract to get rid of the[]s)

l = n.L/Y - r/YSince

lis a whole number andr/Yis strictly between -1 and 1,lmust ben.L/Yeither rounded up or rounded down. Q.E.D.The sign of

rdetermines 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. Alsor/Yis the estimateaof how much the new year runs ahead against the seasons. So the remaindery.L mod Yfor any yearyis an estimate of the earliness of next year's new year.

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

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