11–20 of 41 posts
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+...
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.
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+...
No? 434437¹⁵ + 588129¹⁵ = 352114612798389918432959038931874707477944791745072116429171623348641601872653349744942 588544¹⁵ = 352114612798389919037509191076162751408804961099244071160111294752887772695040228327424 It’s apparently close enough to fool Google, but it’s not correct.
[0] https://www.wolframalpha.com/input/?i=588129%5E15+%2B+434437...
[1] https://www.wolframalpha.com/input/?i=(+588129%5E15+%2B+4344...
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.
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.
The CDC 6600 was a pretty hardcore machine; it was Seymour Cray's first design at his Chippewa Falls skunk works, according to Wikipedia. Would definitely be interesting to know how long their solution took.
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.
Once you have the numbers, it's easily reproducible.
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. How was the programming done? How long time did the execution take? The instance is claimed to be "the smallest". In what norm? How can you be sure that the computer program did not skip any smaller examples?
The most serious problem with the paper, however, is that the only contribution is a purely computational statement which, though a counterexample to a conjecture, does not advance mathematical theory in any way.
Due to these shortcomings, my recommendation to the editor is to REJECT the submission.
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.
Earlier quoted context omitted.
No? 434437¹⁵ + 588129¹⁵ = 352114612798389918432959038931874707477944791745072116429171623348641601872653349744942 588544¹⁵ = 352114612798389919037509191076162751408804961099244071160111294752887772695040228327424 It’s apparently close enough to fool Google, but it’s not correct.
~10^-18 difference as a fraction of the latter power [0], but still a substantially huge number (on the order of 10^68) [1]. [0] https://www.wolframalpha.com/input/?i=588129%5E15+%2B+434437... [1] https://www.wolframalpha.com/input/?i=(+588129%5E15+%2B+4344...