Counterexample to Euler's conjecture on sums of like powers
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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
#2[1966]
(Also a nice early-ish use of supercomputers to solve pure math problems).
Re: Counterexample to Euler's conjecture on sums of like powers
#327^5 + 34^5 + 110^5 + 133^5 = 144^5
Re: Counterexample to Euler's conjecture on sums of like powers
#4This 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.
Re: Counterexample to Euler's conjecture on sums of like powers
#527^5 + 34^5 + 110^5 + 133^5 = 144^5
You miscopied; it's 84, not 34.
Re: Counterexample to Euler's conjecture on sums of like powers
#6Is it known what the smallest power is for which three summands are insufficient? Is there a smallest power? I imagine that would be hugely difficult, but for all I know it falls to the same methods as Fermat. (I'm no expert.)
Edit: I guess the answer might be "5" depending on what the paper means by "smallest" example. (smallest largest number vs smallest smallest number.)
Re: Counterexample to Euler's conjecture on sums of like powers
#7[deleted]
Re: Counterexample to Euler's conjecture on sums of like powers
#8Is it known what the smallest power is for which three summands are insufficient? Is there a smallest power? I imagine that would be hugely difficult, but for all I know it falls to the same methods as Fermat. (I'm no expert.) Edit: I guess the answer might be "5" depending on what the paper means by "smallest" example. (smallest largest number vs smallest smallest number.)
95800^4 + 217519^4 + 414560^4 = 422481^4
Re: Counterexample to Euler's conjecture on sums of like powers
#9The 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+...
Re: Counterexample to Euler's conjecture on sums of like powers
#10The 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.