In episode 17 of season 11 of the Big Bang Theory: “The Athenaeum Allocation” Sheldon (a character in the show) describes a wedding date (the 12th of May) as romantic because it has a specific property. In this post I’ll see what other dates in a given year have the same property. This will use the python library SymPy which is a great computer algebra system.
Sheldon describes the date enthusiastically by saying:
“The month squared equals the square of the sum of the members of the set of prime factors of the day”
If a given date has day \(D\) and month \(M\) this mathematically implies:
So for example for the 12th of May, we have \(M=5\) (May is the 5th month) and \(D=12\):
We see that actually the “squared” part of the condition is redundant. This is because if:
but as everything here is positive integers we can just omit this all together. So Sheldon should have said:
“The month equals the sum of the members of the set of prime factors of the day”
(I’ll give the show that this is perhaps less artistically impressive)
So what other days could this marriage date land on?
I’m going to investigate this using Python and the fantastic SymPy library which will quickly get us the prime factors of a number.
>>> import sympy as sym >>> sym.primefactors(12) [2, 3]
We’re using Python which is a language not just used in science so has many useful tools so first of all let’s write a generator that gets all dates in a year:
import datetime as dt def dates_in_year(year): """ Generator that yields all dates in a given year """ date = dt.date(year, 1, 1) while date.year == year: yield date date += dt.timedelta(1)
We can then combine this with the number theoretic abilities of SymPy to get all dates that follow the condition:
def find_dates(year): """ Generator that yields dates in a given year for which the sum of the prime factors of the date is equal to the month. """ for date in dates_in_year(year): day, month = date.day, date.month if sum(sym.primefactors(day)) == month: yield date
Now we can print out all these dates:
>>> count = 0 >>> for date in find_dates(year=2017): ... print(date) ... count += 1 2017-02-02 2017-02-04 2017-02-08 2017-02-16 2017-03-03 2017-03-09 2017-03-27 2017-05-05 2017-05-06 2017-05-12 2017-05-18 2017-05-24 2017-05-25 2017-07-07 2017-07-10 2017-07-20 2017-08-15 2017-09-14 2017-09-28 2017-10-21 2017-10-30 2017-11-11 >>> print("Count:", count) Count: 22
We see that there are 22 dates (including the 12th of May) that follow the condition in 2017.
Usually, weddings are on weekends so which of those dates happens to fall on a weekend?
def find_weekend_dates(year): """ Generator that yields weekend dates in a given year for which the sum of the prime factors of a date is equal to the month. """ for date in find_dates(year): if date.weekday() >= 5: yield date
>>> for date in find_weekend_dates(year=2017): ... print(date) 2017-02-04 2017-05-06 2017-10-21 2017-11-11
So we see that actually the 12th of May is not an “OK” date in 2017.
Indeed, it looks like it’s a Friday in 2017:
>>> dt.date(2017, 5, 12).weekday() 4
Let us check 2018:
>>> for date in find_weekend_dates(year=2018): ... print(date) 2018-02-04 2018-03-03 2018-05-05 2018-05-06 2018-05-12 2018-07-07 2018-10-21 2018-11-11
Note that apart from the 12th of May and the 21st of October, these are “trivial”, they are dates where the \(M\)th day of month \(M\) is on a weekend and is also prime.
If this wedding was to take place in 2018 (North hemisphere) Summertime it looks like the 7th of July is the date.