
Andrew Booker Solves Sum-of-Three-Cubes Problem for 33 - dnetesn
http://nautil.us/issue/70/variables/how-search-algorithms-are-changing-the-course-of-mathematics
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yorwba
Same topic 3 days ago:
[https://news.ycombinator.com/item?id=19492091](https://news.ycombinator.com/item?id=19492091)

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xaedes
Algorithm is explained in the video:

    
    
      x³ + y³ + z³ = 33
      (x+y)(x²-xy+y²)=33-z³
      d=x+y
      x²-xy+y²=(33-z³)/d
    

Guess values for d, because of integer constraint z³ can only take certain
values for a specific d. Try them all. Now that z and d are fixed only two
unknowns x & y remain with the last two equations to solve from.

    
    
      x+y     =d
      x²-xy+y²=(33-z³)/d

~~~
superpermutat0r
This might be constrained even more by other identities.

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jhncls
Extra tricks to further constrain the search and some implementation details
can be found in Andrew Booker's paper [0]. Hacker News of a few days ago
contains additional thoughts [1].

[0]
[https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf](https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf)

[1]
[https://news.ycombinator.com/item?id=19492091](https://news.ycombinator.com/item?id=19492091)

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jhncls
Does anybody have an idea what they mean by the article's title? This kind of
extensive numerical search with suitable constraints has been going on since
the dawn computers. Are these algorithms changing the course of mathematics
for the last 50+ years?

~~~
philipov
Does the 4 color theorem count?

The problem with brute force proof is that proofs are supposed to bring
understanding. They aren't supposed to just assert that something is true, but
also explain _why_ it is true, and by doing so, they open the door for further
research. For this reason, a proof is often more valuable than the result
itself.

The proof of Fermat's Last Theorem is important because it exposes deep
connections between previously disconnected disciplines, not because the
identity itself is particularly useful. On the other hand, brute force proofs
are inscrutable and offer none of that.

While Goedelian Incompleteness garuantees that not everything can be
understood this way, that is not an excuse to use brute force more than
absolutely necessary.

~~~
pyrale
The 4 color theorem brought a novel way of creating proofs. It may not help
much with the field of graph theory, but it contributed to other fields of
knowledge.

