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Perpetual Calendar

By David A. Smith

This topic allows us to compute the day of the week given any date after a in the Gregorian calendar. Recall, that a leap day occurs on February 29th for those years (leap years) that are divisible by 4 except those that are divisible by 100, but including those years that are divisible by 400. In particular, this topic also illustrates how to setup conventions and how to use modular arithmetic to compute the day of the week for a given year. Several examples are given for birthdays of famous mathematicians.   

Example (Perpetual Calendar Conventions) Using the Gregorian calendar, let the days of the week be represented by the following numbers:  



and let the months of a year be represented by the numbers:



Using this convention we can express each date (after 1600) using the following variables:  



For example, for the date October 31, 1996, we have and For the date, January 22, 1970, we have and Also, for the date January 4, 1800, we have and

Proposition (Perpetual Calendar) Using the Gregorian calendar, let the days of the week be represented by the following numbers:  



and let the months of a year be represented by the numbers:



Using this convention we can express each date (after 1600) using the following variables:  



Then the day of the week of day of month of year is given by



    Proof. The proof is divided into two parts. (i) First we develop a formula to determine the day of the week of March 1 for any given year. (ii) Then we will extend this formula to cover of any day of any month of any year for the Gregorian calendar.
    (i) It is known that March 1, 1600 was on a Wednesday. Thus,
    
        March 1, 1601 fell on day
        March 1, 1602 fell on day
         March 1, 1603 fell on day
          March 1, 1604 fell on day
      
In general, March 1 of the year falls on day number where



where is the number of elapsed leap years. The function is given by



where is given by

Therefore, the formula to determine , the day on which March 1 falls in the year is thus







Because, we see and it is easy to show that and so





Therefore, the formula to determine , the day on which March 1 falls in the year is thus



    For part (ii), we will now extend this formula to cover other dates, we consider the following information: There are zero days from March 1 to March 1, there are days from March 1 to April 1 and there are days from March 1 to May 1 and , ... , there are days from March 1 to February 1, and In summary we have,  


    
In order to attain a formula we must take into account this information; that is a formula that inputs the number of the month and outputs the day of the week of first day of the given month as listed in the above tables. Fortunately, Rev. Zeller has determined that the function:  

defined for 1, 2 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12

takes on the same values (mod 7). For example, and  

    

Now using the formula for the day on which March 1 falls in the year which given by



we simply add days to this to obtain the first day of the month for any given month for any given year ,  



Finally, to determine the day of the week for any day , of any given month , of any given year , we have,



and simplified we have,



as desired.

Example (Perpetual Calendar) Determine the day of the week for the date:
    
(David Hilbert's Birthday) January 23rd, 1862

    Solution.  Since, and we have,





Therefore, the day of the week is Thursday.
    
Example (Perpetual Calendar) Determine the day of the week for the date:

(Georg Cantor's Birthday) March 3rd, 1845

    Solution.  Since, and we have,





Therefore, the day of the week is Monday.  
    
Example (Perpetual Calendar) Determine the day of the week for the date:

(Gertrude Mary Cox's Birthday) January 13th, 1900

    Solution. Since, and we have,





Therefore, the day of the week is Saturday.  

Example (Perpetual Calendar) Determine the day of the week for the date:

(Albert Einstein's Birthday) March 14th, 1879

    Solution.  Since, and we have,





Therefore, the day of the week is Friday.  
    
Example (Perpetual Calendar) Determine the day of the week for the date:

(Johann Carl Friedrich Gauss's Birthday) April 30th, 1777

    Solution.  Since, and we have,




    
Therefore, the day of the week is Wednesday.

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