Happy Ramanujan's Birthday

By the way, as I write this it is Srinivasa Ramanujan's birthday (December 22).

Ramanujan, as not-nearly-enough people know, was an amazingly intuitive, brilliant mathematician, whose tragically short life makes a remarkable story. One anecdote that illustrates the kind of observation he would just casually throw off now and then concerns the number 1729, now known as Ramanujan's number, which happened to be the number of a taxicab his friend and fellow mathematician G. H. Hardy took to visit him in the hospital one day. Apparently the pair were fond of working out "interesting" facts about numbers they randomly encountered, because Hardy mentioned 1729 to Ramanujan and added that he hadn't been able to think of anything particularly interesting about it.

Ramanujan at once countered that it was, in fact, a very interesting number, being the smallest number that is the sum of two squares in two different ways.

His insight was correct, as it often was (though not always): 1729 is 10 cubed plus 9 cubed (1000 + 729), and also 12 cubed plus 1 cubed (1728 + 1).

I actually make very brief passing reference to this anecdote in a story of mine that has been accepted into one of the upcoming Superversive Planetary anthologies, which I'll be writing more about later.

Meanwhile, here is a slightly curious fact relating Ramanujan's number to the coming new year, 2019, that AFAIK no one in the world is aware of, except me, and now you (if you're the first one to read this post, and to get this far... which is pretty likely). If you take the 14th root of 1729, and add 5, and raise the result to the 4th power, you get VERY close to 2019. Within about one-one hundredth, as I recall from when I worked it out. Plugging 1729 into this works better than either 1728 or 1730, too.

It also happens that 1729 to the power 6/7 (that is, raised to the sixth power and then extracting the seventh root) is quite close to the whole number 596.

Another original-with-me curiosity of modest interest: If you raise 12 to the 3rd power and add one, then of course you get 1729. But if you add one to 12 and then raise to the 3rd power (that is, take the cube of 13), you get 2197, which is a palindrome of 1729. There's another number smaller than 12 that has this property, that if you cube-and-add-one or add-one-and-cube you get two numbers that are palindromes: can you find it? As I recall there are several more such numbers, at least one of them below 100, and a few more below 2000.

Ah, well. Enjoy Ramanujan's Birthday!