>>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.
> 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.
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!
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.
*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. ).
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:
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.
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.