I watched a movie yesterday and highly recommend it. The Man Who Knew Infinity is based on the true story of the mathematician Srinivasa Ramanujan. I don’t want to say too much for fear of giving it away, but the movie takes us through the development of Ramanujan’s mathematical career. He was a brilliant, self-taught man whose work is still being pored over. Although the applications of some of his discoveries were not apparent at his time, his insights are being applied today to the study of black holes. The movie does have an abrupt, sad ending, which I should have seen coming. Once I shook off enough of the depressed feelings from the end, I could think for a while on one of the mathematical truths in the movie.
Really, this mathematical truth is one for which G.H. Hardy and Ramanujan are quotably famous. G.H. Hardy had gone to visit Ramanujan in the hospital, and he remarked to Ramanujan that his cab number of 1729 was a dull one and hopefully not an omen. Ramanujan replied, “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” I had heard this quote many times before the movie, but I hadn’t taken the number 1729 and played with it. So, I did so for a little while yesterday evening. I made a list of the cubes from 1 to 1,728. I stopped there, because a larger cube would take me beyond 1,729. I then examined my list to see which ones would add up to 1,729. I quickly found the sums 729+1000 and 1+1728.
The process was not difficult, but it did leave me to ponder how Ramanujan had come to his conclusion. Was it really as simple as that? The movie indicated that he was brilliantly fast when it came to computations in his head. He said that he would pray to his family god and then have the results appear in his head. But, what was really happening? Did he just recently work with this number 1,729 so that it was fresh in his mind, or did he do an analysis that quickly in his interaction with Hardy? Either way, it’s impressive.
I went on to stare a while longer at the numbers. The prime factorization of 1,729 is 7 x 13 x 19. I thought it interesting that 13-7=6 and 19-13=6. And, if you factor the sum of cubes, you end up with either 19×91 or 13×133. The digits of the first number are found in the first and last digits of the second number. Likely a coincidence but still fun to ponder. After this, I grew tired and quit thinking on 1,729. I believe that I will revisit it sometime soon and will post when I do.