If numbers can get along, why can’t we? We can work it out!

Several years ago, I read the delightful book The Housekeeper and the Professor by Yoko Ogawa.  The story wasn’t exciting, but it was relaxing and enjoyable.  I was impressed by how well the author worked real mathematics into the plot.  I didn’t pick up the book again after completion, as I am not one for reading books or watching movies multiple times.  There’s too much out there, and life’s too short.  Anyhow, I was reminded yesterday that there was a movie made in Japan that stuck closely to the story.  It is called The Professor and His Beloved Equation, and you can find it on YouTube.  You may need subtitles since it is in Japanese.  I watched it in the afternoon, and I was once again struck by how well the story presents some mathematics.  The mathematical concepts are very clearly explained, and they are at a level that most casual viewers can follow.  And I don’t feel like the math was forced in for the sake of including math.  The film flowed naturally.  I do hope that you will take the time to watch, but I want to go ahead and present some math spoilers this week.  A few of the concepts are too good to not write about them.

Let’s start with something called amicable numbers.  Take the numbers 220 and 284 and find the proper divisors (all divisors excluding the number itself).  Go ahead, I will give you a minute.  (Don’t peek until you are done!)

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

284:  1, 2, 4, 71, 142

Now, find the sum of the proper divisors for each number.  (Again, don’t peek until you are done!)

The sum for 220 is 284.

The sum for 284 is 220.

Wow!  The numbers 220 and 284 are an amicable pair.  The sum of the proper divisors of one number is the other number!

This particular pair was discovered by Nicolo Paganini in 1866.  He was only 16 years old at the time!  He was not the first to discover amicable pairs, but he was the first to find this small pair.  Long before Paganini, Arab mathematicians had discovered amicable pairs, some of which were discovered again later in the 17th and 18th centuries by Fermat, Descartes, and Euler.  I encourage you to do some searching.  Other amicable pairs are larger numbers, which are harder to discover but can still be fun to verify.

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