Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Counterexample to Euler's conjecture on sums of like powers

fermatslibrary.com

Post reply on HN
Print view View original

31–40 of 41 posts

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

#31

Post by ipsin »

vostok wrote: post #29
kmiroslav wrote: post #25

Earlier quoted context omitted.

> 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.

It advances mathematics by saving the time of people who might otherwise try to prove a false conjecture?

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

#32

Post by reubenmorais »

andrewstuart2 wrote: post #21
waterhouse wrote: post #19

Earlier quoted context omitted.

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. :-)

I believe this is the quote you're referring to, by Erik Demaine on 6.006:

"I use BFS a lot, myself, to check mathematical conjectures. [...] You think something is true, you can imagine some graph of all possible inputs to that theorem and you'd need to check them. Typical way of doing that is BFS through that entire graph of states. Usually we're testing finite special cases of a general conjecture, but if we find a counterexample, we're done, don't have to work on it anymore. If we don't find a counterexample then we have to do the mathematics."

https://www.youtube.com/watch?v=s-CYnVz-uh4&t=8m51s

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

#33

Post by Someone »

CamperBob2 wrote: post #16
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.

As a first attempt, I guess you'd create a table of 5th powers of integers through (say) 1000, then look for combinations of two, three, or four entries from the same array that added up to one of the same numbers. Or more likely, walk through the array and subtract all combinations of one, two, three, or four elements from each, looking for zeroes. The most naive search through such a table could take on the order o…

The naive way:

- compute S0 = {n^5 | 1 - compute S1 = {n^5 | 1 - compute S2 = {s+t | s,t ∈ S1} (500,500 additions)

- compute S4 = {s+t | s,t ∈ S2} (around 10^11 additions)

- compute the intersection of S0 with 4

You don't have to store S4, so 2 megabytes of memory or so should suffice. 10^11 operations likely was out of reach for the CDC600, though.

To speed up things, do the above for numbers up to 100 or so using modulo arithmetic. That gives you candidate solutions modulo your word size. That way, you don't have to compute large parts of S4.

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

#34

Post by [deleted] »

GFK_of_xmaspast wrote: post #12
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.

One could probably start with the fact that the last 2 digits of any 5th power are in "0, 1, 7, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99" and go from there.

[deleted]

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

#35

Post by GnarfGnarf »

CamperBob2 wrote: post #16
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.

As a first attempt, I guess you'd create a table of 5th powers of integers through (say) 1000, then look for combinations of two, three, or four entries from the same array that added up to one of the same numbers. Or more likely, walk through the array and subtract all combinations of one, two, three, or four elements from each, looking for zeroes. The most naive search through such a table could take on the order o…

I've actually written FORTRAN on a CDC 6600, at Dalhousie University. The algorithm was probably written in assembler (COMPASS) for speed. The 6600 had a 60-bit register, advanced for its time. The cabinets had some real cool glass panels. The console was two futuristic vector CRTs.

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

#36

Post by aaron695 »

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…

One is welcome to suggest improvements but to reject on that they owe more is BS anti science, it's politics.

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

#37

Post by amelius »

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…

> No reference is made to previous attempts to study Euler's conjecture or similar conjectures

Why should this be even necessary? There are surveys for that. Not every mathematician needs to be a historian. Nor do most people like to read the same introduction in every paper on a certain topic.

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

#38

Post by sijoe »

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…

While I acknowledge the humor intended, it is important to note that nipping incorrect conjectures before people waste time building edifices atop them, is an extremely good outcome. In the strongest possible manner, this advances mathematics by labeling incorrect pathways as being incorrect.

You only need one counterexample to a theory or conjecture to prove it wrong.

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

#39

Post by Natsu »

sillysaurus3 wrote: post #9

The recent counterexample to Fermat's Last Theorem was pretty interesting, too. 434437^15 + 588129^15 = 588544^15 https://www.google.com/search?q=588129%5E15+%2B+434437%5E15+...

Wasn't that the (fake) one shown on Futurama? Or am I confused?

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

#40

Post by gshubert17 »

Someone wrote: post #33
CamperBob2 wrote: post #16

Earlier quoted context omitted.

As a first attempt, I guess you'd create a table of 5th powers of integers through (say) 1000, then look for combinations of two, three, or four entries from the same array that added up to one of the same numbers. Or more likely, walk through the array and subtract all combinations of one, two, three, or four elements from each, looking for zeroes. The most naive search through such a table could take on the order o…

The naive way: - compute S0 = {n^5 | 1 - compute S1 = {n^5 | 1 - compute S2 = {s+t | s,t ∈ S1} (500,500 additions) - compute S4 = {s+t | s,t ∈ S2} (around 10^11 additions) - compute the intersection of S0 with 4 You don't have to store S4, so 2 megabytes of memory or so should suffice. 10^11 operations likely was out of reach for the CDC600, though. To speed up things, do the above for numbers up to 100 or so using m…

I computed an array of fifth powers, up to 200, then had four nested loops. Using python3 on a 2.66-GHz Core 2 Duo, it took 48 seconds to find the reported numbers. If there had been no solutions, it would have taken only a couple of seconds longer.
Post reply on HN