Hardy, Ramanujan and Taxi No. 1729
johncarlosbaez.wordpress.com>>I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
The quote above is from G. H. Hardy himself, from the book "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work". There was no need for him to embellish the story while it was published to "cheer up" Ramanujan, since the book was published in 1940 after Ramanujan's death.
Two great men can have different interests in the same field. It does not mean one of them had less ability. Hardy, since his early days, was fascinated by pure mathematics and rigor. Ramanujan was playing with numbers on pieces of paper since he was a child. That's why their contributions and intuitions, even though in the same broad field, are so different.
I once correctly guessed a that friend's PIN was "1729", based only on the fact that he was a maths major, a huge fan of Ramanujan, and was sure to have read this story. I still cherish the look of complete confusion on his face, more than 20 years later.
Reading this comment makes me glad I use a different (but still nerdy) pin :)
Is it in here? 4242 9801 1089 1024 3141 1415 2718 1412 1732 1428 2997 6626 6022
1337?
One approach to PINs is to take two numbers that are familiar, but aren't related to each other and interleave them.
Basically the same principle as correct horse battery staple.
> Hardy either knew of Ramanujan’s work on this problem or noticed himself that 1729 had a special property. He wanted to cheer up his dear friend Ramanujan, who was lying deathly ill in the hospital. So he played the fool by walking in and saying that 1729 was “rather dull”.
If this is the case, it really increases my respect for Hardy. Anybody can brag, but to willingly seem to be the fool, in order to help someone else (and notice how even in his retelling of the story, he still plays the fool for others as foil to Ramanujan) takes a really big person.
At one point, people were saying there were 3 great English mathematicians alive: Hardy, Littlewood, and Hardy-Littlewood.
Interesting idea indeed: Him pretending to not know that 1729 isn't a 'dull' number at all. I head the same idea as the authors wife: he said it on purpose!
> it is the smallest number expressible as the sum of two cubes in two different ways.
I don't understand why satisfying this completely arbitrary condition makes it interesting. That way, you can have any number be interesting.
1730 is the smallest number greater than the smallest number expressible as the sum of two cubes in two different ways.
This condition was anything but arbitrary in the context of Hardy and Ramanujan's research at the time, and their work has branched and bloomed into several current research areas. To understand why, I need to give a bit of history.
In the 17th Century, Albert Girard and Pierre de Fermat found exactly which numbers can be represented as a sum of two squares, and gave a formula for the number of such combinations [0].
This result by itself was just a curiosity; what made it interesting and enduring was the depth of the mathematics it spawned. Leonhard Euler and others in the 18th Century studied the natural generalization of the problem by replacing the two squares with the more general quadratic form x^2 + n*y^2 for various integers n. This turned out to be extremely difficult and fruitful and led to 19th and 20th Century class field theory ([1],[2]).
With the theory of representations as quadratic forms being an active research area in the early 20th Century (in which Hardy, Littlewood and Ramanujan contributed significantly), they branched out to higher order polynomials such as sums of cubes, and shifted their focus from the existence of combinations to the function of the number of combinations; that is, from a question of structure to a question of distribution*. This perspective led to numerous developments building connections with harmonic analysis, additive combinatorics, even dynamical systems and ergodic theory. Research in these questions continues to this day.
[0] https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_... [1] http://www.math.toronto.edu/~ila/Cox-Primes_of_the_form_x2+n... [2] Introduction to the Construction of Class Fields, Harvey Cohn.
*For the sum of cubes, in particular, the counting function is mostly a 0-1 function whose distribution was studied by C. Hooley (On the representation of a number as a sum of two cubes, Math. Z. 82 (1963), 259–266) and T. Wooley (Sums of two cubes, International Mathematics Research Notices, 1995(4), 181. ).
Indeed, that's exactly why there's no uninteresting number!
This method is implicitly measuring the interestingness of a number by the number of symbols needed to specify it.
So "interesting" is a measure, not a boolean.
The ratio of the actual Kolmogorov complexity to the superficially apparent Kolmogorov complexity?
By the way...Wikipedia has something on a "naive" attempt to compute the Kolmogorov complexity of a string. It says that iterating through all strings won't work because some of them contain infinite loops and the halting problem is uncomputable.
Ok, but what if you make a program to test all possible strings in "parallel", that is, on a sequential processor, but using brief time slices? That way, shouldn't you finish with all the strings that halt without letting the infinite loops hold things up?
I probably don't understand something or am plagiarizing something I've forgotten, or both.
It’s a long con by the computer scientists who can’t be bothered to constrain the domains of functions beyond int, even though we know square(x: int) -> square_int.
They wave their hands and say everything is interesting.
>> in his “second notebook”. This is one of three notebooks Ramanujan left behind after his death—
Hope the present day Mathematicians, biologist etc still use physical notebooks or non-propretary format note taking apps that will make their work accessible to others after their death.
Ramanujan always amazes me. Remembering my visit to the Ramanujan Museum in India, which treasures the pictures, letters, and documents focusing the greatest mathematician of the 20th Century:
https://casualwalker.com/museum-for-the-man-who-knew-infinit...
This virtual expo is quite nice to have a look, thanks for posting!
I just looked into this story, and there was more to it than we usually hear.
This is a fascinating question. This is a famous story, as a Cambridge mathematician I've heard it many times but the simple question, 'did the cab reg numbers in uk in 1919 allow for a match with ".*1.*7.*2.*9.*"' is not easily answerable.
Number plates at the time were in the format AB1234, so it's possible by "taxi number" he meant the number on the number plate
edit: further research shows there would have been a "license number" too, which could have been 4 digits