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).

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

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

;)

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

"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.

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

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

(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)

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