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Perpetual Calendar

By David A. Smith

Be Happy Do Number Theory 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:  

perpetual calendar _gr_1.gif]

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

perpetual calendar _gr_2.gif]

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

perpetual calendar _gr_3.gif]

For example, for the date October 31, 1996, we have perpetual calendar _gr_4.gif] perpetual calendar _gr_5.gif] perpetual calendar _gr_6.gif] perpetual calendar _gr_7.gif] and perpetual calendar _gr_8.gif] For the date, January 22, 1970, we have perpetual calendar _gr_9.gif] perpetual calendar _gr_10.gif] perpetual calendar _gr_11.gif] perpetual calendar _gr_12.gif] and perpetual calendar _gr_13.gif] Also, for the date January 4, 1800, we have perpetual calendar _gr_14.gif] perpetual calendar _gr_15.gif] perpetual calendar _gr_16.gif] perpetual calendar _gr_17.gif] and perpetual calendar _gr_18.gif] perpetual calendar _gr_19.gif]

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

perpetual calendar _gr_20.gif]

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

perpetual calendar _gr_21.gif]

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

perpetual calendar _gr_22.gif]

Then the day of the week perpetual calendar _gr_23.gif] of day perpetual calendar _gr_24.gif] of month perpetual calendar _gr_25.gif] of year perpetual calendar _gr_26.gif] is given by

perpetual calendar _gr_27.gif]

    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 perpetual calendar _gr_28.gif]
        March 1, 1602 fell on day perpetual calendar _gr_29.gif]
         March 1, 1603 fell on day perpetual calendar _gr_30.gif]
          March 1, 1604 fell on day perpetual calendar _gr_31.gif]
      
In general, March 1 of the year perpetual calendar _gr_32.gif] falls on day number perpetual calendar _gr_33.gif] where

perpetual calendar _gr_34.gif]

where perpetual calendar _gr_35.gif] is the number of elapsed leap years. The function perpetual calendar _gr_36.gif] is given by

perpetual calendar _gr_37.gif]

where perpetual calendar _gr_38.gif] is given by perpetual calendar _gr_39.gif]

Therefore, the formula to determine perpetual calendar _gr_40.gif], the day on which March 1 falls in the year perpetual calendar _gr_41.gif] is thus

perpetual calendar _gr_42.gif]

perpetual calendar _gr_43.gif]

perpetual calendar _gr_44.gif]

Because, perpetual calendar _gr_45.gif] we see perpetual calendar _gr_46.gif] and it is easy to show that perpetual calendar _gr_47.gif] and so

perpetual calendar _gr_48.gif]

perpetual calendar _gr_49.gif]

Therefore, the formula to determine perpetual calendar _gr_50.gif], the day on which March 1 falls in the year perpetual calendar _gr_51.gif] is thus

perpetual calendar _gr_52.gif]

    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 perpetual calendar _gr_53.gif] days from March 1 to April 1 and perpetual calendar _gr_54.gif] there are perpetual calendar _gr_55.gif] days from March 1 to May 1 and perpetual calendar _gr_56.gif], ... , there are perpetual calendar _gr_57.gif] days from March 1 to February 1, and perpetual calendar _gr_58.gif] In summary we have,  

perpetual calendar _gr_59.gif]
    
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:  

perpetual calendar _gr_60.gif] defined for perpetual calendar _gr_61.gif] 1, 2 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12

takes on the same values (mod 7). For example, perpetual calendar _gr_62.gif] and  

perpetual calendar _gr_63.gif]    

Now using the formula for the day perpetual calendar _gr_64.gif] on which March 1 falls in the year perpetual calendar _gr_65.gif] which given by

perpetual calendar _gr_66.gif]

we simply add perpetual calendar _gr_67.gif] days to this to obtain perpetual calendar _gr_68.gif] the first day of the month for any given month perpetual calendar _gr_69.gif] for any given year perpetual calendar _gr_70.gif],  

perpetual calendar _gr_71.gif]

Finally, to determine the day of the week for any day perpetual calendar _gr_72.gif], of any given month perpetual calendar _gr_73.gif], of any given year perpetual calendar _gr_74.gif], we have,

perpetual calendar _gr_75.gif]

and simplified we have,

perpetual calendar _gr_76.gif]

as desired. perpetual calendar _gr_77.gif]

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

    Solution.  Since, perpetual calendar _gr_78.gif] perpetual calendar _gr_79.gif] perpetual calendar _gr_80.gif] and perpetual calendar _gr_81.gif] we have,

perpetual calendar _gr_82.gif]

perpetual calendar _gr_83.gif]

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, perpetual calendar _gr_84.gif] perpetual calendar _gr_85.gif] perpetual calendar _gr_86.gif] and perpetual calendar _gr_87.gif] we have,

perpetual calendar _gr_88.gif]

perpetual calendar _gr_89.gif]

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, perpetual calendar _gr_90.gif] perpetual calendar _gr_91.gif] perpetual calendar _gr_92.gif] and perpetual calendar _gr_93.gif] we have,

perpetual calendar _gr_94.gif]

perpetual calendar _gr_95.gif]

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, perpetual calendar _gr_96.gif] perpetual calendar _gr_97.gif] perpetual calendar _gr_98.gif] and perpetual calendar _gr_99.gif] we have,

perpetual calendar _gr_100.gif]

perpetual calendar _gr_101.gif]

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, perpetual calendar _gr_102.gif] perpetual calendar _gr_103.gif] perpetual calendar _gr_104.gif] and perpetual calendar _gr_105.gif] we have,

perpetual calendar _gr_106.gif]

perpetual calendar _gr_107.gif]
    
Therefore, the day of the week is Wednesday. perpetual calendar _gr_108.gif]

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