Ramanujan Surprises Again (2015) | Hacker News Reader

Ramanujan Surprises Again (2015)

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454 pointstmbsundar92 comments

An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: https://hal.archives-ouvertes.fr/hal-02070778. It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number.

(Technically, it works for any number of dimensions >= 1729, but the proof fails for dimensions less than that. Future work might bring the bound down, or better explain why that bound is necessary.)

In fact, there seems to be a lot of interesting things about 1729: https://en.wikipedia.org/wiki/1729_(number)

Well, now I know why my Scheme class in uni was called CSE 1729.

I love how the article starts with the most boring facts about 1729:

> 1729 is the natural number following 1728 and preceding 1730.

Heh. I've been reading HN for long enough to never be surprised by the capability of incredibly pedantic people to be incredibly pedantic.

We had to have a home somewhere :) And this is it.

If a fact is the most boring[0], though, certainly that's something notable and therefore interesting about it.

Therefore, these are not the most boring facts about 1729.

;)

[0] https://en.wikipedia.org/wiki/Interesting_number_paradox

Well, that shows there is no most boring fact, but I think it doesn't preclude a the existence of a set of most boring facts, with none more boring than all the others. Yes, that is an interesting feature of the facts in the set, but does it mean that they are all necessarily more boring than every fact outside the set?

Another fun fact: 1 + 7 + 2 + 9 = 19 ; 19 × 91 = 1729

This kind of coincidence is just cute, it doesn't imply anything useful mathematically right?

I would not be surprised if it is found to be useful. My reason for this is the Quran's mathematical composition, founded on the number 19: https://www.masjidtucson.org/quran/miracle/

That's numerological nonsense like Bible Code.

It is not; I invite you to study it before you reject it.

The burden of proof is on the person making the claim. Pray supply something more substantial than a link (to what looks like a religious organisation) in support of your claim.

I agree, hence the link. You are asking me to write out in a comment the information that is presented clearly and accessibly in the link.

Took a quick look. All that it seems to say is a version of "19, therefore {deity}didit". Not clear, smells like numerological nonsense, so no thank you, moving on.

The fun thing about math (and science and technology as well) is that it is you can't always tell what is going to useful down the road.

"Interestingness" is often as good a heuristic as any when looking for paths that lead to useful developments, although the path is often not a straight one or short one.

I also like the idea of secondary and tertiary effects. One simple example: By "playing" with cute yet fun ideas that are highly likely to not lead to anything immediately interesting, we can build up skillsets and capabilities that lead to very useful results for other problems. Perhaps this is somewhat akin to how the young of predator species "play" around in a way that prepares them to actually hunt when they are older.

Have any developments come out of adding the digits? My impulse is to dismiss it out of hand because it only works in base-10, which in my mind leans it towards numerology instead of math.

It looks like there are some applications. Checksum algorithms are probably the easiest to appreciate.

https://en.wikipedia.org/wiki/Digit_sum#Applications

I think small random uses add up to practical value although there are those who make a religion out of its 'meaning'.

some (n mod 9) can be found by (is congruent to) (sum of the digits mod 9) instead is the most obvious example

True only in base 10 although similar congruences exist for other bases and also involve adding the digits

I'm terrible at discrete math so I'm not going to work it out, but I'm sure it would be possible to express that relationship in a way that it could adapt to other number bases, and then it might be easier to tell if there is something there.

(something like treating the digits as coefficients of a series of powers of the number base, defining a function f(b,n) that returns the sum of the digits of n in base b, setting that equal to the product of the forward and reverse representation of that result in base b, and seeing if the equation looks interesting)

E.g. you can use a similar technique to show why the finger counting method of multiplying by 9 works, or why multiples of 3 have digits that sum to a multiple of 3 (same for 9)

And it belongs to the first pair > 1000 to both have Wikipedia pages!

It doesn't appear to be anything fundamental, just a number that makes the proof technique work out. As the authors point out, that dimension is only a condition to get some loose bound working, not a necessary condition generally:

It is possible to improve the factor K = 1728 appearing in Corollary 5.5, at the expense of introducing various technical complications into the algorithm. In this section we outline a number of such modifications that together reduce the constant to K = 8 + ϵ, so that the modified algorithm achieves M(n) = O(n log n) for any d ≥ 9 (rather than d ≥ 1729).

Interesting but as it seems currently without practical implications for multiplication:

https://en.m.wikipedia.org/wiki/Galactic_algorithm

He credited his work to his family goddess. From wikipedia:

"A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess. "An equation for me has no meaning," he once said, "unless it expresses a thought of God.""

Mathematics are the best expression of the transcendental divine. Pythagoras and Plato had the same perspective.

Yes, it looks like for many mathematicians, the confusion between stable conceptual foundation and eternal objective reality is too seductive to not fall in the illusion of identity of things locally indistinguishables.

Funny to use the word transcendental there, since the Pythagoreans held ratios to be divine but couldn't figure out irrational numbers, like pi. They had trouble squaring that circle.

That's why I think Pythagoras refused to write down his doctrines. He knew there was more to be empirically discovered -- and he was wary of how text could become dogma. The divine he uncovered was based on a mathematical, harmonious cosmos; but he recognized it was beyond understanding in a lifetime. That's why Pythagorean mysticism is compatible with modern science -- he didn't write anything down!

2000 years later, Kepler had faith in a harmonious cosmos, and charged his model of harmony so it could fit the evidence. He elipsed the circles, instead of squaring them.

Fun fact #1: it is impossible to square a circle [1]

Fun fact #2: the Pythagoreans conducted the first attested scientific experiment in Western history (according to a recent PhD thesis at UMich [2])

[1] https://en.m.wikipedia.org/wiki/Squaring_the_circle

[2] https://deepblue.lib.umich.edu/handle/2027.42/150050

"Transcendental" also appears in Calculus.

Not a mathematician, but it was Thomas Paine's view that the hand of the divine can be seen by the study of nature - which I believe was a view shared by Isaac Newton.

Thomas Paine wrote on his point of view in "The Age of Reason" [0]:

> Each of those churches show certain books, which they call revelation, or the word of God. The Jews say, that their word of God was given by God to Moses, face to face; the Christians say, that their word of God came by divine inspiration: and the Turks say, that their word of God (the Koran) was brought by an angel from Heaven. Each of those churches accuse the other of unbelief; and for my own part, I disbelieve them all.

> As it is necessary to affix right ideas to words, I will, before I proceed further into the subject, offer some other observations on the word revelation. Revelation, when applied to religion, means something communicated immediately from God to man.

> No one will deny or dispute the power of the Almighty to make such a communication, if he pleases. But admitting, for the sake of a case, that something has been revealed to a certain person, and not revealed to any other person, it is revelation to that person only. When he tells it to a second person, a second to a third, a third to a fourth, and so on, it ceases to be a revelation to all those persons. It is revelation to the first person only, and hearsay to every other, and consequently they are not obliged to believe it.

> It is a contradiction in terms and ideas, to call anything a revelation that comes to us at second-hand, either verbally or in writing. Revelation is necessarily limited to the first communication- after this, it is only an account of something which that person says was a revelation made to him; and though he may find himself obliged to believe it, it cannot be incumbent on me to believe it in the same manner; for it was not a revelation made to me, and I have only his word for it that it was made to him.

> ...

> But some, perhaps, will say: Are we to have no word of God- no revelation? I answer, Yes; there is a word of God; there is a revelation.

> THE WORD OF GOD IS THE CREATION WE BEHOLD and it is in this word, which no human invention can counterfeit or alter, that God speaketh universally to man.

> …

> It is only in the CREATION that all our ideas and conceptions of a word of God can unite. The Creation speaketh an universal language, independently of human speech or human language, multiplied and various as they may be. It is an ever-existing original, which every man can read. It cannot be forged; it cannot be counterfeited; it cannot be lost; it cannot be altered; it cannot be suppressed. It does not depend upon the will of man whether it shall be published or not; it publishes itself from one end of the earth to the other. It preaches to all nations and to all worlds; and this word of God reveals to man all that is necessary for man to know of God.

> Do we want to contemplate his power? We see it in the immensity of the Creation. Do we want to contemplate his wisdom? We see it in the unchangeable order by which the incomprehensible whole is governed! Do we want to contemplate his munificence? We see it in the abundance with which he fills the earth. Do we want to contemplate his mercy? We see it in his not withholding that abundance even from the unthankful. In fine, do we want to know what God is? Search not the book called the Scripture, which any human hand might make, but the Scripture called the Creation.

> The only idea man can affix to the name of God is that of a first cause, the cause of all things. And incomprehensible and difficult as it is for a man to conceive what a first cause is, he arrives at the belief of it from the tenfold greater difficulty of disbelieving it. It is difficult beyond description to conceive that space can have no end; but it is more difficult to conceive an end. It is difficult beyond the power of man to conceive an eternal duration of what we call time; but it is more impossible to conceive a time when there shall be no time.

> In like manner of reasoning, everything we behold carries in itself the internal evidence that it did not make itself Every man is an evidence to himself that he did not make himself; neither could his father make himself, nor his grandfather, nor any of his race; neither could any tree, plant, or animal make itself; and it is the conviction arising from this evidence that carries us on, as it were, by necessity to the belief of a first cause eternally existing, of a nature totally different to any material existence we know of, and by the power of which all things exist; and this first cause man calls God.

> ...

> That which is now called natural philosophy, embracing the whole circle of science, of which astronomy occupies the chief place, is the study of the works of God, and of the power and wisdom of God in his works, and is the true theology.

---

[0]: https://www.deism.com/theageofreason.htm

Thanks for sharing.

Isaac Newton: "All my discoveries have been made in answer to prayer."

People forget how religious newton was and he believed his physics was the discovery of god's physical laws.

Chemistry comes from mystic alchemy. Astronomy derives from astrology.

Just like there is a thin line between genius and madness, the same seems to apply to science and mysticism. Turn the dial a few degrees, you get mysticism. Turn it a few more degrees, you get science.

History of science is just as fascinating as history of politics. It's a shame we focus on the latter so much.

There are no real historical conflict between mysticism and science. Take Russel's essay "Mysticism and logic" for example, Russel having evolved from a more Platonist point of view to a more atheist materialist one over its long lifespan, inventing "neutral monism" on the road.

However there are clear possibility of conflicts between scientific approach and institutional religions. Note that people don't need to be anything close to an atheist to get into trouble when their thoughts are going out of the road promoted by the institutional dogma of the day. Giordano Bruno is one famous case of such a human drama.

Of course religion here is more the mean of political control than anything else, and it can be substituted by other means just as detrimental to independent critical free thought.

I think the clear border is that if you start to make things up that go against what we solidly measured so far or you have gaps in your derivation then you are well within 'mysticism' part of the dial. I think there's even slightly audible click when you go to that part of the dial.

I think they believe that every thought that comes to them is the part of common consciousness of the cosmos.

Ramanujan always surprises.

Recently visited the "Ramanujan Museum":

* Where we could see his family goddess - Sri Namakkal Namagiri Thayaar who he credited for his works:

http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...

* His desk where he made his early mathematical findings:

http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...

* Really sadden to see how a Ramanujan's wife lived her life after Ramanujan's demise:

http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...

* Check the full photo coverage about Ramanujan and his fascinating works at:

http://casualwalker.com/museum-for-the-man-who-knew-infinity...

Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. It's interesting to find out that the real coincidence here is that Hardy rode in a taxicab whose number had happened to show up in Ramanujan's investigations of Fermat's last theorem.

A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked.

That threw me for a loop and I started believing shit like no one's smarter than I was etc. Then I just ... grew up, I guess. And I remembered this story by Feynman and I realised that despite his absolutely undoubtable genius, he'd have appeared godlike to me if I was his classmate back in the day.

Ramanujan's brain worked even faster by most accounts. He dreamed in math, I think. So there are multiple stories where people ask him a puzzle and he'll answer with an equation that solves it for the entire family of problems that the puzzle could come from.

What was the quote about Feynman? That he loved to cultivate anecdotes about himself or something similar? Makes a lot of his stories make a lot more sense, too.

It's a rant by fellow Physicist Gell-Mann.

I think it's pretty plausible that he was a raging egomaniac (narcissist, perhaps?).

Undoubtedly a great thinker and genius, but that doesn't say very much about personality traits.

Feynman even has his own 1729 anecdote: https://www.ee.ryerson.ca/~elf/abacus/feynman.html

From Murray Gell-Mann: https://www.youtube.com/watch?v=rnMsgxIIQEE

Several clips from Gell-Mann's Web of Stories interview (late 1990s) pertain to his on-again off-again collaboration with Feynman.

https://www.youtube.com/watch?v=o2sEW4ggVlA&list=PLVV0r6CmEs...

I think the part that makes it genuine is that he was comically self aware of himself and his craziness. Even when he was pushing the boundaries just for the sake of it and to make a caricature / character out of himself, he did it in a way that made me think that he didn't really pretend to not be doing it for his ego.

It's like 4 levels of thinking somehow merged in his actions: 1) be normal and look at the crazy people, 2) be a crazy person, 3) be a crazy person and be aware of your craziness, 4) be a crazy person, be aware of it and let others know that you're aware of it. It feels like one of those thought spirals I go into if I have weed. It's right on the boundary of crazy but probably also (in his case) inside the realm of genius.

i recall him explaining several shortcuts one can use to solve problems in seemingly impossible speeds by drawing on a breadth of experience from similar problems that you have memorized or are easy to compute and interpolating.

its still genius but not in the sense of actually being able to do huge calculations in ones head the way a computer would.

Feynman has no dearth of stories showing his genius, but one specific example is a video 3Blue1Brown did of how Feynman converted the velocity vectors of a revolving body in a gravitational well (I think it was that) into a perfect circle of vectors. It's one of those results that's deep and yet you can find it yourself too if you spend enough time with it.

Often it's also simply just that people are not used to thinking about more efficient ways of solving a problem.

There's a (quite possibly apocryphal) story about Niels Henrik Abel in primary school, where his teacher supposedly wanted time to do some grading and assigned the students the busywork of adding up all numbers from 1 to 100. Abel supposedly quickly found the well known formula n(n+1)/2 and gave the teacher the answer within minutes, and the teacher supposedly believed he'd somehow "cheated" because he could not imagine any of them could figure it out.

I have no idea if the story is real (I grew up in Norway, so Abel was a popular subject for stories like this) - it was told to me in high school by a maths teacher after giving us the modified task of seeing if we could find any shortcuts to doing the sums, and seeing what we'd come up with. I found the formula quickly, but at that age that's nothing special, especially not when prompted to find an alternative solution.

But the overall idea the teacher was trying to get us to understand was how to pause and think about how to decompose a problem rather than just picking the most obvious alternative, and learning to be "lazy" in the sense of relentlessly looking for an easier way to do things is a large part of what got me into software development..

When I heard this story it was about Gauss.

And I looked it up- Yes, the same possibly apocryphal story is on his Wikipedia page: https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss

Interesting. Not surprised this is the kind of story people might have adapted rather freely to sound more familiar to a local audience...

This reminds me of the von Neumann fly puzzle story:

https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abi...

Or swallow, if you think Wigner's account is more accurate than Halmos'.

Yeah, Neumann was well into the nutty end of the genius spectrum. As in, his genius was so nuts that it's hard to believe he was real. Similar with "The strangest man to ever visit my lab" - Paul Dirac (that's a quote from Neils Bohr). At least Dirac appeared to be autistic enough from the description that I'm not surprised he was able to do what he did. It's a superior intelligence, but I'm guessing he was obsessive about shit as well.

Of course, once you know how to sum infinite geometric series it's not too hard to come across the answer that way ;)

I think most of us are impressed by computational parlor tricks (and indeed raw computational intelligence in general -- being able to process information and compute quickly and accurately), but for me, genius goes beyond that.

Genius is about having rare and useful insights that the rest of us are incapable of, and that a computer is unable to easily replicate.

For instance, there was this thing on Twitter recently about all percentages being reversible (7% of 50 is equal to 50% of 7, but the latter is easier to mentally calculate). Most of us are aware that multiplication is commutative, but it takes genius to recognize and frame that insight in a useful way.

It never occurred to me that 7% of 50 is equal to 50% of 7 perhaps because they are equally easy to calculate? Multiply 7 by 5 and fix the decimal point.

What about 17% of 50? 17*5 isn’t so simple anymore, whereas 50% of 17 is 8.5.

To be fair, with most shortcuts, it’s possible to construct difficult cases (17% of 23 is difficult in either order) but where it applies (when one of the pairs is a common percentage), exploiting commutativity can be quite useful. Plus the mental overhead of remembering the rule is extremely minimal.

How can you discover that 50% of 17 is 8.5 if you can’t multiply 17*5? (If the answer is “by halving”, then my response is that halving and then multiplying by 10 is often the easiest way to multiply by 5!)

I’m not sure, but to me at least halving 17 to get 8.5 is an almost completely intuitive process, whereas multiplying 17 by 5 seems to require an extra cognitive step to reduce it to shifting the decimal point and then halving (or vice versa). Never mind the extra step when presented by the problem of taking 17% of 50.

I'm not that sure about computational intelligence being all that impressive. For instance, that parlour trick you mentioned is common enough to come across (I'm not particularly a super genius but I had it figured out by the time I was in college) that I was surprised that so many people found it useful. Like legit intelligent people were talking about how useful it was on Twitter. Which made me realise that some of us were just quicker with math even if our overall intelligence wasn't spectacular.

That's not an example of genius by any stretch of imagination, sorry.

Feynman was undoubtedly a genius, but he also suffered from a need to be admired. The safecracking episodes at Los Alamos are a perfect example - giving the impression he was an expert safe cracker when his real methodology was guesswork and sometimes subterfuge (birthdays, anniversaries, or even subtlety observing someone inputting their combination).

I get your point, I really do. But if what you call his "real methodology" worked well, why would he use more advanced/safecracker-y techniques?

I'd say the real mistake he made was that he lifted the veil off of how he did things, leading people to say "oh even I could have done that".

Something something, invoice $1 for chalk, $9999 for knowing where to mark the 'X'.

zigackly (https://farm5.static.flickr.com/4096/4746820383_314f398f2f.j...)

David Kelly [1] once told me that as a grad student at Princeton, he somehow managed to get the Sunday New York Times delivered to him late Saturday night. He'd stay up all night solving the crossword puzzle, then dazzle everybody who was stumped on it the next day.

[1] https://www.vinc17.net/yp17/index.en.html

I'm not sure I'd call it coincidence rather "being struck by the vast amount of time and passion Ramanujan put into mathematics". I'd be absolutely amazed if he couldn't have recalled a similarly obscure fact for most numbers below 10,000.

> Ramanujan's innate calculating capability.

Hard work and obsessive work effort on a specific area makes it appear innate.

> When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability.

This is the way the story is always presented, and I think that's usually how it's intended, but I think it's quite misleading for another reason too. If you've ever made or looked at a table of cubes, the famous fact really jumps out (in base 10). I'm serious, look:

The two pairs of cubes are 1000 and 729, and 1728 and 1, and 1000 and 1 make the addition trivial and the similarity obvious (and 729 and 1000 are even right next to each other, one row away from 1728!). With that observation, it doesn't take much effort to try the smaller possibilities and see that 1729 is the smallest number that can be written as the sum of two cubes two different ways. Ramanujan knew numbers and their relationships intimately, better than Hardy, who knew more theory. I think Ramanujan knew the fact about 1729 already, and that you are right about the taxi number coincidence being more surprising and, well, impressive.

(Yes, I've commented on this before: https://news.ycombinator.com/item?id=21165031)

The taxi cab story is easily a top-5 math story, and is quintessential Ramanujan.

Has there been a genius of his kind since? Maybe Terry Tao, but his work also lacks the ease and lack of machinery that Ramanujan had. Truly amazing.

Yes definitely: Alexandre Grothendieck. And Terrence Tao can’t sit at his table (yet?).

But honestly it’s kind of a silly game to rank mathematicians this way.

Grothendieck and lack of machinery do not belong in the same sentence. But yes, it is kind of silly.

> Grothendieck and lack of machinery do not belong in the same sentence.

They most certainly do! Grothendieck’s work is heavy on definitions, but the essence of his work is that the right definitions obviate (and are seen to be right because they obviate) the need for heavy machinery. See the famous quote, taken from Wikipedia (https://en.wikiquote.org/wiki/Alexander_Grothendieck#Quotes_...) because that’s the first place I found it:

> I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.

I mean, we're arguing semantics at this point, but these definitions we're talking about are things like schemes, which pretty much everybody would call heavy machinery.

There have been lots. Twenty? Thirty? A hundred? It was a good century plus for mathematical genius.

Has there been a genius of his kind since? Maybe Terry Tao, but his work also lacks the ease and lack of machinery that Ramanujan had. Truly amazing.

It's hard to compare mathematicians, but I suppose Erdős[1] would be in the conversation.

[1]: https://en.wikipedia.org/wiki/Paul_Erd%C5%91s

What are the other 4 top math stories?

For me one of them has to be of Évariste Galois[1], who, legend has it, hastily wrote fragments of his last mathematical discoveries on his shirt sleeves before fighting the duel that would end his life.

[1] - https://en.wikipedia.org/wiki/%C3%89variste_Galois

Fun mathematical tourism stop: go up to the top of the Tour Montparnasse. Look straight down. Galois is buried in the cemetery below. Nobody knows where, but he's down there somewhere.

My understanding is that this legend has been debunked, but ironically I can't find a source. (My instinct is to blame E. T. Bell.)

I am surprised no one in including Gauss.

Didn't Gauss live and die before Ramanujan was born?

John von Neumann is someone that often comes up in this context - there are numerous anecdotes on how he was perceived as frightingly clever; also his body of work is beyond impressive.

Discussed at the time: https://news.ycombinator.com/item?id=10518452

I always found Ramanujan very intriguing. He operates on a dimension that is unknown to most of us. Makes me wonder if he is a great yogi or a time traveler.

Fantastic article that explains the math (and physics) very clearly.

Interesting read, The title should have 2015 in it though.

Added. Thanks!

Ramanujan also claimed 1 + 2 + 3 + ... = -1/12.

How does that work? Who can explain this to me?

Here's a Numberphile video on this

https://www.youtube.com/watch?v=w-I6XTVZXww

I think this is basically a “shock value” interpretation of a more subtle statement. Obviously adding strictly-positive numbers does not result in a negative number under normal arithmetic. Check the numberphile video and you may be simultaneously irritated and disappointed.

Burkard Polster's (Mathologer) videos on the subject are probably going to be more useful than Numberphile. Numberphile merely presents a trick; Mathologer points out both that it's patent nonsense [0] and that it's useful patent nonsense [1].

[0] https://www.youtube.com/watch?v=YuIIjLr6vUA [1] https://www.youtube.com/watch?v=jcKRGpMiVTw

Don't we think that the credit for the number 1729 should belong to Hardy, for he took the cab and mentioned that number to Ramanujan. Of course, Ramanujan could see beauty in every number and would have produced something equally beautiful for some other number Hardy could utter.

Reply on news.ycombinator.com