Deriving While Driving

Today is the birthday of Max Zorn born in 1906 in Krefeld, Germany. Zorn is responsible for Zorn's lemma which states that if a partial ordered set exists and any subset of that set has an upper bound then the original set has a maximum element.

The quote of the day is by Srinivasa Ramanujan. Replying to G.H Hardy's suggestion that the number of a taxi (1729) was "dull", "No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being (1 + 1728) and (729 and 1000)."

A week ago, our school held its commencement for the class of 2015. During the week prior to graduation, I spent a great deal of time with the seniors that were in my class and in the organizations that I advised. Our conversations were reflective on their growth and their excited and nervous anticipation of their upcoming educational journeys. Each class attempts to leave a legacy their senior year and some classes do have a more resilient legacy to the corrosiveness of time.

Each numerical year seems to have its own identity. My high school graduation year, 1976, was the bicentennial year. The year 1984, was wrapped in the Orwellian concept of Big Brother. 1999 was made famous by the singer, Prince. The year 2000 was the millennial year. 2008 had the 8 rotated so that it became the infinity symbol. Members of the class of 2012 asked the question, "Don't you want to be 1  2?" 2013 ends in 3 and is also divisible by 3! 2015 was the year of π because on March 14 at 9:26 am, the first eight digits of that irrational number occurred. 

I was contemplating these numerical wonderments at a stoplight on my way to school the day after the commencement ceremony and was drawn to the class of 2016. In my meandering thoughts, I jumped to the factorization of the 2016 and was shocked in my determination that 2016 has 36 factors. Factors are natural numbers that 2016 can be divided by without any remainder. The factors of 2016 are: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016. I started to wonder about the number of factors in each of the years that I have taught. The chart below depicts those graduation years and the number of their factors.

To determine the number of factors, I found the prime factorization of each year. The prime factorization of 2016 would be 2 x 2 x 2 x  2x 2 x 3 x 3 x 7 or in exponential form, 2^5 x 3^2 x 7^1. I used the exponents in the exponential form to find the total factors that exist in 2016. The total number factors can be found by using the following arithmetic: (5 + 1) x (2 + 1) x (1 + 1). Using my high school graduation year, 1976, I found that its prime factorization is 2^3 x 13^1 x 19^1. The exponents are 3, 1, and 1. The number of factors in 1976 is 16; [(3 +1) x (1 + 1) x (1 + 1)].

When I examined the list, I noticed there are five prime years, 1987, 1993, 1997, 2003, and 2011. I believe the next graduating class that will have exactly 36 factors will be the class of 3168. The prime factorization for 3168 is 2^5 x 3^2 x 11^1. The last graduating class to have 36 factors was the class of 1800, 2^3 x 3^2 x 5^2. I was not teaching in 1800 and am not planning on teaching in 3168!