[nneonneo]

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.)

[pishpash]

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).

[user2994cb]

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

[pixelpoet]

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

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

[russellbeattie]

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

[JoeAltmaier]

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

[wwweston]

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

[mkl]

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?

[nexuist]

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

[8bitsrule]

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

[taberiand]

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

[dwaltrip]

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.

[Cpoll]

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.

[dwaltrip]

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

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

[mafaa]

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

[nitrogen]

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)

[BGthaOG]

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/

[lonelappde]

That's numerological nonsense like Bible Code.

[BGthaOG]

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

[emmelaich]

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

[sllabres]

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

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