Ladies and Gentlemen, the next number on the runway is truly perfect!

In the previous post, we looked at numbers that are amicable.  Along a similar line of thinking, let’s look at perfect numbers.  A small example of a perfect number is 6.  What makes it perfect?  Once again, I am going to ask you to take a minute and find the proper positive divisors of 6.  Go on!  I’ll give you time.  (Don’t peek until you are done!)

6:  1, 2, 3

Now, like we did previously, find the sum of these proper divisors.

1 + 2 + 3 = 6

Ta da!

The proper divisors of 6 add up to 6.  Cool!

Any positive integer that has proper divisors adding up to itself is a perfect number.  Can you think of others?  The first 4 were found over 2,000 years ago.  Current findings use a similar line of thinking to that for finding Mersenne primes.  Turns out that perfect numbers and Mersenne primes are closely related.  I will share more on that in the next post.  Until then, happy hunting!

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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.

The New MRI: Faster Than A ?

In the summer of 2012, my young son had a brain MRI to check for a chiari malformation.  (He has a long medical history, due to inheriting a craniofacial syndrome from me.)  After having been through several CT scans in his short life, I was accustomed to worries about sedation so that he would be perfectly still for the imaging.  I was very surprised when we arrived at the hospital, for they did not need to sedate, and the scan was not going to take very long.  He needed the short version.  (The long version would have still required sedation.)  We went through the scan and waited about an hour for the results.  Thankfully, there was no chiari malformation.  He did have to stay in the hospital, as we were also there for ICP (intracranial pressure) monitoring and a skull surgery.  He “just” needed a bone graft and not a full cranial-vault remodeling, and we now knew this from the MRI results.

In the months and year that followed, I didn’t think too much about the speed of the MRI, until a colleague came into my office and shared about Terry Tao and compressed sensing.  Terry Tao is a person who appears frequently in mathematical news, as he is the top mathematician of our age.  Typically, a mathematician picks one small area in which to be an expert and focuses on that for a career.  Terry Tao is rare in that he can pick up a topic of interest, quickly become an expert, and then make large contributions to that area of mathematics.  This just gives you a hint of why my colleague was sharing about Terry Tao.  Anyhow, I have digressed and must get back to compressed sensing.

What is compressed sensing, and how was it possibly related to my son’s MRI?  Compressed sensing involves an algorithm that utilizes l1 minimization.  It takes too much time and storage space to collect every pixel of data for an image.  So, a camera or other device collects a fraction of these pixels.  The l1 algorithm starts arbitrarily picking effective ways of filling in the missing pixels.  Then, the algorithm starts putting in layers of colored shapes over the randomly chosen pixels while seeking sparsity.  It wants to use the simplest kinds of shapes to closely match the existing pixels.  Each new layer will have smaller and smaller shapes.  Eventually, with enough layers, the resulting image will be extremely close to the original.  The simplest or sparsest image is the closest to the original.

So, collecting the data during the MRI scan does not take as long, as fewer pixels are needed.  Processing takes a little while, because the l1 algorithm needs time to work.  I don’t know if this discovery was in place for my son’s MRI, but this story did give me something to relate to.  His scan took only a few minutes, and we had to wait about an hour to see the results.  All but one picture was clear, and we were relieved that he does not have a chiari malformation.

I am eager to see what all developments will come out of this discovery.  Emmanuel Candes, Justin Romberg, and Terry Tao have laid groundwork and proven mathematically that the resulting image after running the l1 algorithm will be extremely close to the original.  Now, people are looking at all kinds of applications.  Besides constructing medical images quickly, people may restore old files, construct images of space, eliminate the need for compression software, and accomplish much more with speed and accuracy.

If you would like to read more, here are 3 articles that helped me to gain better understanding.

Fill in the Blanks: Using Math to Turn Lo-Res Datasets into High-Res Samples

Better Math Could Make Medical Diagnostics 6X Faster

Compressed Sensing Makes Every Pixel Count

Jim Simons: Renaissance Mathematician

Jim Simons is a man who has lived a full life as a mathematician.  While doing mathematical research and laying the foundation for string theory, he also cracked codes for the NSA.  He then went into cracking Wall Street, applying mathematics to investments and becoming wealthy.  He is now active with his Simons Foundation, which supports math and science education and autism research.  Watch this TED talk to find out more.

A Rare Interview with the Mathematician who Cracked Wall Street

NOVA “The Great Math Mystery”

“Mathematics is the alphabet with which God wrote the universe.” – Galileo Galilei

This NOVA special is an entertaining exploration of whether mathematics is a product of human discovery or invention.  An attempt to find an answer goes through history and spotlights the contributions of names such as Fibonacci, Pythagoras, Plato, Galileo, Newton, Maxwell, Marconi, Englert, and Higgs.

The Great Math Mystery

Architectural Ode to Calculus

Did you know that writing a math book can make you a millionaire?  Well, it can if you are James Stewart.  He wrote a calculus text that was so well-done that it continues to be a big seller, even after his death.

If you want to find out more about his texts, check out his site.

Stewart Calculus Books

What is even more interesting is that Stewart’s home was an architectural ode to calculus.  Called Integral House, it has many beautiful curves.  If you have an extra $26.6 million lying about, then this house could be yours.

Integral House for Sale