© J R Stockton, ≥ 2012-05-07

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Here,
unless otherwise indicated, Easter is Gregorian.

From Table III in the Prayer Book, the range of the Gregorian PFM is the 29 days March 21st to April 18th. That matches the Julian range, by design.

Therefore, Easter Sundays, being one to seven days later, are always in March 22nd to April 25th inclusive, on the corresponding Calendar - 35 possible ordinary dates.

The code in Easter Date Frequencies would report any date outside the expected range.

Gregorian Easter will have the next latest possible date on 2038-04-25 (previous: 1943); and the next earliest possible date on 2285-03-22 (previous: 1818). Easter 2008 was early, on 2008-03-23 (previous: 1913, next: 2160). Easter 2011 was late, on 2011-04-24 (previous: 1859, next: 2095).

Julian Easter, respectively : 2078 (1983), 2010 (1915), 2037 (1953), 2051 (1888).

Using ISO 8601
Week Number dates,
the date of Easter Sunday will clearly be of the form
`yyyy-Www-7`.

Extreme Easter Sundays Year Those require a 2285-03-22 = 2285-W12-7 Normal range of six ISO 2972-03-22 = 2972-W12-7 Leap week-numbers, 1943-04-25 = 1942-W16-7 Normal suggesting that no 3784-04-25 = 3784-W17-7 Leap more are needed.

To check over a wide range :-

A full-cycle Gregorian scan from 1583 to 5701582 shows lines just for 1583, 1584, 1585, 1595, and 3784. Easter Sunday can only have the six Week Numbers 12-17 (17 can occur only in a Leap Year, and is rather rare); the counts are 714400, 1330000, 1330000, 1338600, 977250, 9750.

Therefore there are only 6 possible ISO Week-Numbering dates for Easter Sunday.

The same range should apply for Julian Easter on the Julian Calendar.

The Week Number of Easter Sunday is that of Maundy Thursday, three days earlier, and the range of Maundy Thursday is March 19th to April 22nd. Those are Day 78 or 79 of the year, and Day 112 or 113, respectively. Thursday 78 is in Week 12, and Thursday 113 is in Week 17. But there remains the question of whether the latest Easter dates can occur in the appropriate types of Year.

March 22nd can be Day 81 or 82 of the year; April 25th can be Day 115 or 116. Easter Sunday can occur on all four Ordinal Dates; but on Day 081 only on ordinary years and on Day 116 only in Leap years.

Therefore there are 36 possible Ordinal Dates for Easter Sunday (as should be obvious).

I have read that, in the Lunar Calendar, Easter is always the third Sunday in the Paschal Lunar Month.

The Calendar Act does not define that, because it refers only to the Full Moon and not to the start of the Lunar Month.

Clavius, however, considers the beginning of the Lunar Month.

Because the date of Easter is determined by an approximate Moon, and the dates of the true Lunar Month depend somewhat on the observer's location, Easter is sometimes not the third Sunday of the local Lunar Month.

Although the details of the Lunar Month are not necessarily defined by Easter Rules, it is clear that there is a Full Moon one to seven days before each Easter Sunday.

The length of a year is between 12 and 13 lunar months. If an Easter falls earlier in the year than the previous Easter did, then there will have been 12 intervening lunations, otherwise 13.

By so counting (program MJD_DATE, OddTests, Paschal, MeanMoon; and below), I find that 5,700,000 Gregorian years contain 70,499,183 lunar months (confirmed on the Web); that is a prime number, which further confirms that there is no quicker repeat.

The average Gregorian lunar month length, from that, is 29.53058690056025 days or 29d 12h 44m 2.708s. The astronomical value is 29.53059 days or 29d 12h 44m 2.8s. A real difference of 0.1s would require an Ecclesiastical Lunar Leap Day about once per 25 million months.

I find that 6580 Julian lunar months are contained in 532 Julian years, corresponding to the Metonic ratio 235:19. The Julian secular calendar has a period or 28 years; 28 & 19 are co-prime so there can be no faster repeat than 28×19 = 532 years.

The following averages between the Easters of the given years.

The secular Julian and Gregorian Calendars repeat in cycles of 28 and 400 years respectively. But the corresponding dates of Easter Sunday repeat only in cycles which are large multiples of those.

In the old Dionysian (Julian) canon, as the Venerable Bede knew (*A
History of the Englich Church and People*, V.21), the pattern of
Easters repeated every 532 (28×19) years (which is 194313
days).

For Gregorian Easter, the pattern repeats every 5,700,000 years (which is 2081882250 days).

My program longcalc can calculate Gregorian Easter in two ways; years checked for repetition by me : -32500 to >+5,700,000 against 5,700,000 more; and samples enormously further apart. For confirmation, see Calculation of the Ecclesiastical Calendar; Frequency of the Date of Easter over one complete Gregorian Easter Cycle; or a Web search for '+Easter +"5,700,000"'.

Since it agrees with several independent respected methods, one of the functions which I have derived from the Church of England Prayer Book can be used as a starting point.

In the body of that function, taken line-by-line, if YR is
increased by 5,700,000 =
2^{5}×3×5^{5}×19 :-

`GN` is unchanged

`xx` rises by 57000

`CY` rises by ((57000×3/4) - (57000×8/25))%30
= (42750 - 18240)%30 = 24510%30 = 0

`xx` rises by 5700000×(1 + 1/4 - 1/100 + 1/400) =
570×12425 = 7×1011750

`SN` is unchanged

`DM` is unchanged

Therefore, the dates of the Paschal Full Moon and of Easter Sunday
are always unchanged after 5,700,000 years.

They could perhaps repeat more often; but, if so, they would repeat by some factor of 5,700,000 years divided by one or more prime factors. The prime factors are 2, 3, 5, 19. For a starting year, 1943 is good as it had the latest possible, and therefore a rare, Easter date.

So there is no quicker repeat of the entire pattern.

Allowing for the different lengths of the Gregorian and Julian years, the combined Easter date cycle is LCM(365.2425×5700000, 365.25×532) = 1,013,876,655,750 days, which is 2,775,900,000 Gregorian years and 2,775,843,000 Julian years.

Disregarding year length, as the Julian cycle is 19×28 years and the Gregorian cycle is 19×30×25×400 years, the combined cycle should be 7 Gregorian cycles, 39,900,000 years.

Consecutive Easters are always separated by 50, 51, 54, or 55 weeks. Because the lunar month is about 29.5 days, 12 months are about 354 days or 50.5 weeks and 13 months are about 383.5 days or 54.8 weeks; so that should have been expected.

Gregorian tests using a
`longcalc` script covering
over 5,700,000 consecutive years found no counter-examples.

C:\EPHEMERA>longcalc 0 5700000 (eastdiff.scr) SCR ( eastdiff.scr : longcalc script to test that the interval) wrt wln ( between Gregorian Easters is 350, 357, 378 or 385 days) wrt wln ( www.merlyn.demon.co.uk >= 2001-04-17) wrt wln wln (dup wrt wln dup dup #ge swp 0 0 0 #ds swp inc dup #ge swp 0 0 0 #ds ) (swp sub 86400 div dup (DiffDays) wrt wrt 7 div dup wrt ) cat (2 div 26 sub abs dec dup wrt ) cat (((non-zero → not 50 51 54 55 weeks) wrt hlt) no0 ) cat (wln wln) cat 2 kio for stk ( eastdiff.scr ends. ) wrt

This is now confirmed by the code for Individual Easter Date Repeats, which shows that Julian and Gregorian Easter have the same set of intervals.

The average interval is of course 365.2425 days = 52.1775 weeks.

Gregorian Easter occurs with approximately constant frequency on dates from March 28 to April 20, at about once per thirty years (April 19 occurs a little more frequently), with a roughly linear fall-off over a week to the extreme dates March 22 and April 25 - this is unsurprising. Each date occurs a multiple of 25 times in a 5700000 year cycle (Julian Easter has a similar pattern; multiple of 4, in 532 years).

See estr-tbl.txt for frequencies of common dates of Gregorian Easter Sundays in 1900-2199.

The following calculates histograms for Julian and Gregorian Easter Sunday, expressed as common date, ordinal date, and Week Number, for a chosen range of years.

Note that in this 21st century Easter Sunday is most often in Week 15; I have proposed yyyy-W15-7 as a good date for a Fixed Easter Sunday.

A window, initial width zero, slides up the years. At each step, if the window contains too few dates, its front is advanced, otherwise its rear is advanced. Using the date added or removed at each step, counts are maintained for the number of each date within the window, and the number of different dates in the window.

For Julian Easter on the Julian Calendar, I get that the 72-year span AD 1980-2051 includes all 35 dates -MM-DD, and no shorter span does so. Many Russian Orthodox can now hope to complete that span.

For Gregorian Easter on the Gregorian Calendar, I get that AD 1799-1886 = 88 years is the best past span, and AD 6920-6991 = 72 years is the first best span. Had the Bull been earlier, AD 1568-1639 would have been such a span.

The code now also handles Ordinal and ISO Week-Numbering dates. For Ordinal Dates, 36 take 164 Julian and 126 Gregorian years, neither including any part of this 21st century. All six Gregorian Week-Numbering dates need at least seven years, the first such run of dates being 3781-W12-7, 3782-W15-7, 3783-W14-7, 3784-W17-7, 3785-W14-7, 3786-W13-7, 3787-W16-7. Six Julian ones also need seven years.

The code could possibly be adjusted for Julian Easter on the Gregorian Calendar (where, long-term, there are 366 possible common and ordinal dates, and 53 possible ISO 8601 Week-Numbering dates).

See Individual Easter Date Repeats below : For -MM-DD dates - Julian, 247 years; Gregorian; 1887 years.

The second column shows the successive dates at which, including the given year, the number of different Easter Sunday dates seen becomes the Count. The third column similarly shows preceding dates.

The triple entries are respectively for Common Dates
yyyy-mm-dd, Ordinal Dates yyyy-ddd, and Week-Numbering Dates yyyy-Www-d;
see *via* ISO 8601.

Note that for full results, frequencies should be calculated over the full period, but intervals over somewhat longer. Covering the full Gregorian range is slow.

See also Easter Intervals by George W Walker.

Needs checking for Leap and Common. to see when calendars match exactly !!!

N.B. : Easter dates repeat in an irregular manner. As there are only 35 ordinary dates on which Easter Sunday can fall, there cannot possibly be any period of over 35 years which contains no repeated Easter date, in either Calendar.

Many, but not most, Gregorian Easters 84 years apart do match; but Easters 28 or 56 years apart do not match within 1583-9999 at least. As it happens, 1916 and 2000, 84 years apart, were both Leap and both had Easter Sunday on St George's Day (but 0303-04-23 was the Friday after Julian Easter).

An Easter date can repeat after 5 years (*e.g.* 8th
April, 2007-2012); the second year will be Leap. No shorter interval is
possible, since no Sunday date can repeat faster. Dates in March 26th to
April 23rd can repeat after 5 years; it seems that the minimum for March
22nd and April 25th to repeat is 57 years, and for the other four is 11
years.

The largest possible interval between repeats of a specific Easter
date seems to be 1887 years (first from 22nd March 171812); the third
largest, 1651 years, starts on 25th April 106804. The minimum largest
interval is 79 years (first from 19th April 2212). For most dates, the
largest interval is 119 or 147 years. The soonest largest interval, of
141 years, starts on 27th March 2016; the next is for 19th April
(calculations by `mjd_date`).

The possible intervals are : 5 6 11 17 35 40 46 51 57 62 63 68 73 79 84 95 119 125 130 141 147 152 163 179 209 220 231 247 277 288 293 299 304 315 372 383 451 467 524 535 592 603 671 676 687 755 896 907 975 991 1059 1127 1279 1363 1431 1499 1583 1651 1803 1887 years.

Pascal/Delphi DOS-mode
programs paschal, mjd_date, and longcalc (*via* Directory, TXT
and HTML calculate Easter dates;
consider for longcalc
"*DOS>longcalc cof (dup wrt #ge wrt wrt wln) 2000 2020 for*" ;
program envicalc has a script for Gregorian Easter).

???

The possible intervals are : 5 6 11 51 62 73 79 84 95 163 247 years.

N.B. : As Common Dates, but 36.

The possible intervals are : 5 6 11 35 40 46 51 57 62 63 68 73 79 84 95 119 125 130 135 141 147 152 163 179 209 220 231 247 277 288 293 299 304 315 372 383 451 467 524 535 592 603 676 687 755 896 907 975 1059 1127 1279 1363 1583 1594 1746 1898 2118 2270 2490 2574 2642 2710 2794 2862 2946 3014 3082 3712 3864 4084 4168 4236 4304 4456 4540 4608 4676 4760 4828 4912 years.

N.B. : As Common Dates, but 6.

The possible intervals are probably : 2 3 5 6 8 9 11 14 16 19 27 68 84 152 220 288 304 372 524 592 896 3712 3864 4084 4168 4236 4304 4456 4540 4608 4676 4760 4828 4912 years. Note that W17 is rare.

Easter and Easter-linked holidays can match within the range
1980-1999 - of course, Easter Sunday is always on the same day of the
week, so if the Easter date matches, almost everything from the
beginning of March to the following February 28 must also match.
Remember that Gregorian Easter can currently only fall on 35 different
dates (Mar 22 to Apr 25), so there is *a priori* a >50% chance
of any given year having an Easter date match in 1980-1999. I believe
that there are the following Easter matches within 2000-2007 :-
2001=1990; 2002=1991; 2004=1982; 2006=1995.

I have read in Wikipedia Talk:Computus : "The Easters of 1948 to 2047 are repeated from 2100 to 2199, so 100 years on a row. Tom Peters, 23 March 2009"; it is indeed so.

There are, perhaps surprisingly, many long repeated spans.

Gregorian result lines can (currently) be tested visually by pasting
into `F.X0` at User Testing with
code like :-

var IN = F.X0.value.split(/\D+/) var Y1 = +IN[2], Y2 = +IN[3], Shift = +IN[6], A = "", D1, D2 F.Code.rows = 9 ; F.Result.rows = Y2 - Y1 + 6 for (Y = Y1-2 ; Y <= Y2+2 ; Y++) { D1 = DATE2.giveXEasterSunday( Y ) D2 = DATE2.giveXEasterSunday(Y+Shift) A += D1.toString(" Y M D ") + D2.toString(" Y M D ") + " " + (D1.toString("MD") == D2.toString("MD") ) + "\n" }

Selected Gregorian Easter results for spans of at least 100 years
within years 0 to 700,000

Span these years = those years Shift 2123 136490 - 138612 562090 - 564212 425600 2120 48986 - 51105 474586 - 476705 425600 2117 11484 - 13600 437084 - 439200 425600 2113 8991 - 11103 434591 - 436703 425600 2110 6491 - 8600 432091 - 434200 425600 2103 1498 - 3600 427098 - 429200 425600 1101 0 - 1100 425600 - 426700 425600 713 132400 - 133112 414000 - 414712 281600 702 114001 - 114702 258001 - 258702 144000 588 44423 - 45010 188423 - 189010 144000 466 9634 - 10099 153634 - 154099 144000 398 1901 - 2298 145901 - 146298 144000 323 7181 - 7503 61181 - 61503 54000 246 4617 - 4862 58617 - 58862 54000 226 1501 - 1726 55501 - 55726 54000 172 2728 - 2899 21100 - 21271 18372 172 1928 - 2099 20300 - 20471 18372 165 1941 - 2105 8341 - 8505 6400 159 8341 - 8499 20313 - 20471 11972 155 4728 - 4882 23100 - 23254 18372 144 356 - 499 1100 - 1243 744 139 2728 - 2866 14700 - 14838 11972 131 9769 - 9899 16169 - 16299 6400 123 9177 - 9299 9921 - 10043 744 119 7181 - 7299 19153 - 19271 11972 117 8788 - 8904 15188 - 15304 6400 116 4700 - 4815 9984 - 10099 5284 108 6700 - 6807 18384 - 18491 11684 107 9593 - 9699 21565 - 21671 11972 104 2195 - 2298 8595 - 8698 6400 103 2713 - 2815 7997 - 8099 5284 102 1914 - 2015 13598 - 13699 11684 101 1528 - 1628 13500 - 13600 11972 101 1515 - 1615 6799 - 6899 5284 100 2100 - 2199 8348 - 8447 6248 100 1976 - 2075 20500 - 20599 18524 100 1952 - 2051 8200 - 8299 6248 100 1948 - 2047 2100 - 2199 152 * 100 200 - 299 11580 - 11679 11380 100 0 - 99 18372 - 18471 18372

Selected Gregorian Easter results for spans of at least 22 years
around about now matching in years 0 to 250,000

Span these years = those years Shift 398 1901 - 2298 145901 - 146298 144000 358 1941 - 2298 91941 - 92298 90000 102 1914 - 2015 13598 - 13699 11684 172 1928 - 2099 20300 - 20471 18372 165 1941 - 2105 8341 - 8505 6400 100 1948 - 2047 2100 - 2199 152 * 100 1952 - 2051 8200 - 8299 6248 100 1976 - 2075 20500 - 20599 18524 30 1995 - 2024 2215 - 2244 220 74 1995 - 2068 2739 - 2812 744 61 2008 - 2068 2600 - 2660 592 92 2008 - 2099 9000 - 9091 6992 88 2012 - 2099 15100 - 15187 13088 74 2019 - 2092 3135 - 3208 1116 25 2020 - 2044 2392 - 2416 372

The Gregorian Easter pattern repeats every L = 5,700,000 years, the Julian every L = 532 years. Those will not be detected above, because no terminating non-match can be found. For every shift of S, there will necessarily be shifts of N×L ± S.

The largest Julian span is 16 years; 446-461 = 530-545. For spans of four or more years, there is really only one Julian shift, 84 years.

Two main types of match are possible : the two Easters are
simultaneous, *i.e.* on the same physical Day; or the two Easters
are on the same Date in their respective Calendars, *i.e.* their
YYYY-MM-DD match. In a year in the range 200-299, both types of match
could occur at once. In the far future, there will be matches of Day, of
MM-DD date, and of both, with the Gregorian year number greater than the
Julian year number.

Lunar Months | |||
---|---|---|---|

True Length | Julian | Gregorian | |

Rule | (1900) | 235 lm / 19 Jy | less 8 d / 2500 y |

Days | 29.5305882 | 29.5308511 | 29.5305923 |

Error | observed | +0.0002629 | +0.0000041 |

Day out in | n/a | 308 y | 19500 y |

The Gregorian and Julian Easter rules are intended to calculate the
same thing, a luni-solar anniversary. But they do not necessarily give
the same actual day. They usually agree in the First Millennium, agree
about half the time in the Second, and never (I think) agree in the
same-numbered year after AD 2698. That sort of behaviour is
inevitable, because both rules put Easter within a given region of the
calendar year, but the calendar years diverge. Within AD 26-35, for
28 29 31 32 35 the Julian is a week earlier, but for 26 27 30 33 34 they
agree. The difference is always an exact multiple of 7 days, since the
Week is consistent. (*Details in this paragraph depend on my Pascal
implementations of Julian Easter, for which some historical check data
now received (MAK).*)

Matches of Easter Day are apparently most common (as expected) around the 3rd Century, and become steadily less common until the last, in 2698. After about 50 millennia they will recur with differing year numbers (not shown here).

Matches of Calendar Date can only occur in centades where the Gregorian and Julian dates differ by an integer multiple of seven, and in years when the Gregorian and Julian moons are at similar phases.

Only in the 3rd centade can both types of match occur for years of the same number; this will never occur in future.

My program mjd_date can generate file eastdiff.txt, whose format is similar to

Year Gregorian MJD Julian MJD Diff sameDate 2005 G: 3-27 53456 J: 4-18 53491 35 2006 G: 4-16 53841 J: 4-10 53848 7 2007 G: 4-08 54198 J: 3-26 54198 0 2008 G: 3-23 54548 J: 4-14 54583 35 2009 G: 4-12 54933 J: 4-06 54940 7 2010 G: 4-04 55290 J: 3-22 55290 0 2011 G: 4-24 55675 J: 4-11 55675 0 2012 G: 4-08 56025 J: 4-02 56032 7 2013 G: 3-31 56382 J: 4-22 56417 35 2014 G: 4-20 56767 J: 4-07 56767 0 2015 G: 4-05 57117 J: 3-30 57124 7

and

COLS &1581 : < eastdiff.txt | find " 0"

will isolate the years for which Gregorian and Julian Easter are on
the same day. Further use of my program `COLS` can isolate and
count these years; there are 271, starting with 1583, the last being
2698. The calendars differ by three days in 400 years, so after roughly
50 millennia the Easter of one Julian year will occasionally be
simultaneous with that of the next-numbered Gregorian year. I have read,
and confirmed by enhancing program mjd_date, that Julian 44733-04-25 and
Gregorian 44734-03-25 are the first; that series of matches lasts to
Gregorian 47916-04-02; then from 97755-04-06 ....

The `SameDate` column shows `<----` for years when
the Easters are on the same date by their respective calendars. This
last happened for 1298-04-06, and next happens for 5806-04-06; it was
usual in the 3rd centade, will happen throughout c.68, and will be
frequent in c.69.

See also The Date of Easter Sunday