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Counterexample to Euler's conjecture on sums of like powers

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Re: Counterexample to Euler's conjecture on sums of like powers

#21

Post by andrewstuart2 »

waterhouse wrote: post #19
andrewstuart2 wrote: post #15

A fine example of the fact that brute force is often a powerful heuristic. I believe I recall hearing from the MIT OCW Algorithms course that they often rely on computer time doing exhaustive search (i.e. brute force) to "not disprove" some algorithm they develop before sinking too much time into developing a mathematical proof.

I think "exhaustive search" is a misnomer here, as one obviously cannot exhaustively search the integers. (Unless we might say the machine gets exhausted.)

I think the particular quote I'm remembering was in the context of graph algorithms, where exhaustive search is more appropriate a term. :-)

Re: Counterexample to Euler's conjecture on sums of like powers

#22

Post by crypto5 »

fdej wrote: post #18

The submitted paper is well-written and appears to be correct, but it is far too short. No reference is made to previous attempts to study Euler's conjecture or similar conjectures (there are surely a dozen other papers that ought to be cited in the introduction). What led the authors to consider a computer search as opposed to any other approach? Does the method generalize? It is indicated that the CDC 6600 was used…

Lol, as someone who is not in academia and math, I didn't recognize your joke while reading this for the first time.

Re: Counterexample to Euler's conjecture on sums of like powers

#23

Post by JackFr »

fdej wrote: post #18

The submitted paper is well-written and appears to be correct, but it is far too short. No reference is made to previous attempts to study Euler's conjecture or similar conjectures (there are surely a dozen other papers that ought to be cited in the introduction). What led the authors to consider a computer search as opposed to any other approach? Does the method generalize? It is indicated that the CDC 6600 was used…

I can't tell if this is sarcasm or not.

Re: Counterexample to Euler's conjecture on sums of like powers

#24

Post by [deleted] »

andrewstuart2 wrote: post #15

A fine example of the fact that brute force is often a powerful heuristic. I believe I recall hearing from the MIT OCW Algorithms course that they often rely on computer time doing exhaustive search (i.e. brute force) to "not disprove" some algorithm they develop before sinking too much time into developing a mathematical proof.

[deleted]

Re: Counterexample to Euler's conjecture on sums of like powers

#25

Post by kmiroslav »

fdej wrote: post #18

The submitted paper is well-written and appears to be correct, but it is far too short. No reference is made to previous attempts to study Euler's conjecture or similar conjectures (there are surely a dozen other papers that ought to be cited in the introduction). What led the authors to consider a computer search as opposed to any other approach? Does the method generalize? It is indicated that the CDC 6600 was used…

> though a counterexample to a conjecture, does not advance mathematical theory in any way.

[I realize it's sarcasm but in case it's not clear to everyone, I'll speak my mind about it]

I think it's a pretty serious advancement. Confirming a conjecture is just as important as invalidating it with a counter example. It doesn't matter if that counter example fits on one line.

Re: Counterexample to Euler's conjecture on sums of like powers

#26

Post by logicallee »

fdej wrote: post #18

The submitted paper is well-written and appears to be correct, but it is far too short. No reference is made to previous attempts to study Euler's conjecture or similar conjectures (there are surely a dozen other papers that ought to be cited in the introduction). What led the authors to consider a computer search as opposed to any other approach? Does the method generalize? It is indicated that the CDC 6600 was used…

for anyone wondering, the parent is 100% sarcasm and making fun of editors. (The REJECT is the punchline).

You can tell by "How can you be sure that the computer program did not skip any smaller examples?"

However, in actuality it really is actually too short. For example, it would be far more interesting to us in 2016 to read this early computational result if it included "reference made to previous attempts to study Euler's conjecture or similar conjectures", also "What led the authors to consider a computer search as opposed to any other approach?" and so forth.

Also in 2016 it would be extremely interesting to know "How was the programming done? How long time did the execution take?" For purely history-of-computation reasons.

Also it would be really interesting for the authors to have listed their own views on the ramifications, i.e. in what way it advances mathematical theory (or not).

Even though the parent comment is sarcastic, as a reader in 2016, I got next to nothing out of the paper. If it had been rejected on the above (sarcastic) basis, and the authors had submitted a six-page version, it would be far more interesting to us. The same would have been true for contemporaries.

Re: Counterexample to Euler's conjecture on sums of like powers

#27

Post by pajop »

gunnihinn wrote: post #4

This is great from so many angles. The counterexample is solid mathematics, the paper is wonderfully to the point, and the use of computers to successfully investigate a pure math problem was innovative. I only wish they'd described the method they used to find the counterexample.

There's a more detailed discussion at "A Survey of Equal Sums and Power" by L. J. Lander, T. R. Parkin and J. L. Selfridge http://www.jstor.org/stable/2003249 (under 5.1.n)

Re: Counterexample to Euler's conjecture on sums of like powers

#29

Post by vostok »

kmiroslav wrote: post #25
fdej wrote: post #18

The submitted paper is well-written and appears to be correct, but it is far too short. No reference is made to previous attempts to study Euler's conjecture or similar conjectures (there are surely a dozen other papers that ought to be cited in the introduction). What led the authors to consider a computer search as opposed to any other approach? Does the method generalize? It is indicated that the CDC 6600 was used…

> though a counterexample to a conjecture, does not advance mathematical theory in any way. [I realize it's sarcasm but in case it's not clear to everyone, I'll speak my mind about it] I think it's a pretty serious advancement. Confirming a conjecture is just as important as invalidating it with a counter example. It doesn't matter if that counter example fits on one line.

Maybe my mind is warped, but I actually agree with that. The conjecture isn't the point. There are plenty of interesting counterexamples, but a direct search does nothing to advance mathematics.
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