The mathematical formula that reveals when Easter is every year
You can track the start of spring and the phases of the moon—or you can turn to a formula by mathematician Carl Friedrich Gauss
For those who celebrate it, tracking what day the holiday Easter takes place on can be a challenge. According to Christian religious traditions, Easter Sunday falls on the first Sunday following the first full moon after the vernal equinox.
As if that wasn’t confusing enough, not all Christians agree on what day of the calendar that specific Sunday is. Eastern, or Orthodox, Christians, use a Julian calendar, while Western Christians, made up primarily of Roman Catholics and Protestants, use a Gregorian calendar (the calendar used in most parts of the world) to calculate their dates. For ease, I’m focusing my article on the latter group, for whom April 5, 2026, is the day that the Easter Bunny arrives.
How did April 5 get calculated for this year? Well, the vernal equinox, or start of spring, is fixed as March 21. If a full moon occurs on that exact day, March 22 becomes the earliest possible calendar date for Easter Sunday. According to the lunar calendar, the latest possible date for a full moon after March 21 is April 18. That means Easter Sunday never falls later than April 25.
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To calculate the date of Easter for any given year, you have to know not only the distribution of weekdays across calendar dates but also the corresponding lunar phases for each day. But there is a shortcut that German mathematician and astronomer Carl Friedrich Gauss spotted and used to create a formula. Because both lunar phases and days of the week follow fixed rules, the date of Easter can be determined mathematically.
A Simple Easter Formula
In essence, Gauss’s solution is remarkably simple. You merely need the following formula: 22 + d + e.
This sum corresponds to the date in March on which Easter Sunday falls. If the result exceeds 31—for instance, if it sums to 40—then you simply count the excess days into April. (In 2026, for instance, Easter falls on April 5, which corresponds to a sum of 36.)
Sounds simple, doesn’t it? To evaluate the formula, however, you need to know two values, d and e, which both will depend on the specific year for which you wish to determine the date of Easter. And this is where things, unfortunately, become a little more involved.
Strictly speaking, four steps are required to find these values:
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First, three numbers—a, b and c—must be determined. The first, a, is obtained by dividing the year number by 19 and calculating the remainder. (For 2026, the result is a = 12, because 2026 ÷ 19 = 106 with a remainder of 12.) The number b is obtained by dividing the year number by 4 and calculating the remainder. For 2026, the result is b = 2. And c is the remainder obtained by dividing the year number by 7; thus, for 2026, c = 3.
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Next, three additional numbers—k, p and q—are required. To calculate k, you must divide the year number by 100 and keep only the integer part of the result. For 2026, the result is therefore k = 20. The value for p is obtained by dividing k by 3 and keeping only the integer part—resulting in p = 6 for 2026. The final number, q, is obtained by dividing k by 4 and keeping only the integer part; that is, q = 5 for 2026.
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From the numbers determined previously, two additional values—M and d—can now be derived. To find the value of M, calculate 15 + k − p − q, which, for the 2026 example, yields 15 + 20 – 6 – 5 = 24.The value of d is obtained by calculating M + 19 × a,dividing the result by 30 and keeping only theremainder. For the year 2026, d = 12.
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The values N and e are determined in a similar way as M and d. First compute 4 + k − q and divide the result by 7; the remainder is the value of N. For the year 2026, N = 5. The value of e is obtained by calculating 2 ×b + 4 × c + 6 × d + N, dividing the result by 7 and again keeping only the remainder. Thus, for the year 2026, e = 2.
Following these calculation steps, you can determine the values for d and e, which you can use to determine the date of Easter for any given year—with two exceptions. They are:
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If d = 29 and e = 6, then Easter falls on the March 50—that is, April 19.
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If d = 28, e = 6 and a > 10, then Easter falls on the March 49—that is, April 18.
For the year 2026, neither of these two exceptions applies. The date for Easter Sunday can be calculated using the formula mentioned above: 22 + d + e = 22 + 12 + 2 = 36. Because March 36 does not exist, this date corresponds to April 5, 2026.
Even though Gauss’s Easter formula is easy to evaluate, determining the values needed to find d and e can be a bit tedious. Gauss never claimed that his solution was easy to memorize.
Over time, experts have repeatedly attempted to devise simpler Easter formulas. No one has been truly successful in doing so. This is unlikely to surprise anyone who has ever delved into calendar mathematics. And finding the date for Easter has additional complications: it depends not only on calendar days but also on astronomical events such as the full moon and the vernal equinox.
All in all, my advice for finding the date of Easter is to check your analog or digital calendar.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
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