Gaussian weekday formula

The Gaussian weekday formula is a formula by Carl Friedrich Gauß for calculating the weekday of January 1st of any year. Gauss described this method in a handwritten note on a collection of astronomical tables, but never published it himself. It was only published posthumously in his collected works in 1927 .

As an extension of this method for the beginning of the year, there are formulas that calculate the day of the week for any date (see day of the week calculation ). They are partly traced back to Gauss and are also referred to as the Gaussian weekday formula.

Formula for the beginning of the year

The weekday of January 1st of year A is according to Gauss:

{\ displaystyle w = (1 + 5 \ cdot ((A-1) {\ bmod {4}}) + 4 \ cdot ((A-1) {\ bmod {1}} 00) +6 \ cdot (( A-1) {\ bmod {4}} 00)) {\ bmod {7}}}

Here mod denotes the modulo operation. The variables have the following meaning:

A.: The (four-digit) year
w: The weekday. The days of the week are numbered from 0 to 6 with Sunday as the beginning of the week with the day number 0, see the following table.

0	1	2	3	4th	5	6th
Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday

The formula applies from the introduction of the Gregorian calendar on October 15, 1582 and in principle until these calendar rules are deviated again, i.e. until the next calendar reform. The correctness of this formula is not immediately apparent, but has been proven by Berndt Schwerdtfeger.

example

Example: January 1, 2015

     A = 2015
     (A-1) mod 4 = 2
     (A-1) mod 100 = 14
     (A-1) mod 400 = 14
     1 + 5*2 + 4*14 + 6*14 = 151
     151 mod 7 = 4

January 1, 2015 was a Thursday (day number 4).

Extensions for any date

There are several extensions to this method for any date. For example, Kraitchick described a formula that he also traced back to Gauss.

Web links

Find the day of the week of January 1st of each year. Golden number. Epacts. Easter border . In: Carl F. Gauß: Works , Vol. XI i

Individual evidence

↑ Carl F. Gauss: To find the weekday of January 1st of a year. Golden number. Epacts. Easter border . In: Royal Society of Sciences to Göttingen (Hrsg.): Works . 2nd reprint edition. Volume XI i. Georg Olms Verlag, Hildesheim 1981, ISBN 978-3-487-04643-3 , p. 206-207 (reprint).
↑ ^a ^b Berndt E. Schwerdtfeger: Gauss' calendar formula for the day of the week (PDF) May 7, 2010. Retrieved June 29, 2016.
^ Maurice Kraitchik: Chapter five: The calendar . In: Mathematical recreations , 2nd rev. [Dover]. Edition, Dover Publications, Mineola 1942, ISBN 978-0-486-45358-3 , pp. 109-116.

[Gauss-1] Carl F. Gauss: To find the weekday of January 1st of a year. Golden number. Epacts. Easter border . In: Royal Society of Sciences to Göttingen (Hrsg.): Works . 2nd reprint edition. Volume XI i. Georg Olms Verlag, Hildesheim 1981, ISBN 978-3-487-04643-3 , p. 206-207 (reprint).

[Schwerdtfeger-2] Berndt E. Schwerdtfeger: Gauss' calendar formula for the day of the week (PDF) May 7, 2010. Retrieved June 29, 2016.

[Kraitchik-3] Maurice Kraitchik: Chapter five: The calendar . In: Mathematical recreations , 2nd rev. [Dover]. Edition, Dover Publications, Mineola 1942, ISBN 978-0-486-45358-3 , pp. 109-116.