3 ms·

Andrew Booker Solves Sum-of-Three-Cubes Problem for 33

58 points by dnetesn 8y ago

xaedes 8y ago
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 8y ago
  This might be constrained even more by other identities.
  - jhncls 8y ago
    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
jhncls 8y ago
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 8y ago
  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 8y ago
    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.
- dang 8y ago
  That title does seem hyperbolous. We switched to the HTML doc title.
yorwba 8y ago
Same topic 3 days ago: https://news.ycombinator.com/item?id=19492091 https://news.ycombinator.com/item?id=19492091

58 points by dnetesn 8y ago

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 8y ago
This might be constrained even more by other identities.
- jhncls 8y ago
  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

jhncls 8y ago

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 8y ago
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 8y ago
  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.
dang 8y ago
That title does seem hyperbolous. We switched to the HTML doc title.

yorwba 8y ago

Same topic 3 days ago: https://news.ycombinator.com/item?id=19492091 https://news.ycombinator.com/item?id=19492091