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 flouran 2009-12-10 05:36

Diophantine Equation

I recently received an email from some postgraduate student in Moscow (possibly a crank) who asked me the following question:
[QUOTE]
Does the equation x^10+y^10+z^10=t^4 have any solutions in positive
integers?
[/QUOTE]

I thought I would ask this forum if anyone could give either an affirmative or negative answer to this question....

Thanks!

 alpertron 2009-12-10 11:59

I received the same message at least 5 times in my e-mail in several years, but I don't know how to solve it.

 Batalov 2009-12-10 22:32

No solutions with x<=y<=z<=1000.
(Brute-forced; only with a trivial observation that of x,y,z, one will be odd and two other, even; the case of all of them even is reducible).

No solutions above 1000 would be probabilistically expected, I'd think.

 flouran 2009-12-11 03:51

[QUOTE=cmd;198396]Hilbert (10)

L(k,n,x,y,z,...)=R(k,n,x,y,z,...)

no solutions[/QUOTE]

Are you saying then that the reason why x^10+y^10+z^10 = t^4 does not have any solutions in positive integers is a direct consequence of Matiyasevich's Theorem?

I think that the incorporation of certain ingredients of Matiyasevich's proof and a variant of FLT can be used to prove that x^10+y^10+z^10 = t^4 does not have any solutions in positive integers.

 CRGreathouse 2009-12-11 05:53

[QUOTE=flouran;198446]I think that the incorporation of certain ingredients of Matiyasevich's proof and a variant of FLT can be used to prove that x^10+y^10+z^10 = t^4 does not have any solutions in positive integers.[/QUOTE]

I think not.

1. It's not obvious how Matiyasevich's theorem (Hilbert X, Robinson-Davis-Putnum-Matiyasevich, etc.) applies; it doesn't show that there are no solutions, only that proving that you've found all solutions is hard in the general case.
2. This isn't like the general case. Diophantine equations with 9 variables are known to be universal, but only with ridiculously high degrees (~10^45 as I recall). You have a degree-10 equation with only four variables.
3. Wiles' theorem doesn't seem well-equipped for the additive explosion on the LHS. Three terms is vastly different from two terms.

Also, there aren't many mathematicians in the world capable of extending his proof, and (to my knowledge) none here/

 flouran 2009-12-11 06:11

[QUOTE=CRGreathouse;198455]I think not.

1. It's not obvious how Matiyasevich's theorem (Hilbert X, Robinson-Davis-Putnum-Matiyasevich, etc.) applies; it doesn't show that there are no solutions, only that proving that you've found all solutions is hard in the general case.
2. This isn't like the general case. Diophantine equations with 9 variables are known to be universal, but only with ridiculously high degrees (~10^45 as I recall). You have a degree-10 equation with only four variables.
3. Wiles' theorem doesn't seem well-equipped for the additive explosion on the LHS. Three terms is vastly different from two terms.
[/QUOTE]
Thank you for the verification! I appreciate it!
[QUOTE=CRGreathouse;198455]
Also, there aren't many mathematicians in the world capable of extending his proof, and (to my knowledge) none here/[/QUOTE]
Would it be worth asking Wiles or one of his former Ph.D. students (i.e. Brian Conrad)?

 maxal 2009-12-12 06:17

[QUOTE=flouran;198360]I recently received an email from some postgraduate student in Moscow (possibly a crank)[/quote]

I guess that was the same guy who asked similar questions to a number of members of NMBRTHRY mailing list.

Here is some background info:
[url]http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0908&L=nmbrthry&T=0&F=&S=&P=1025[/url]
and his question is actually related to the contest set up by a story
"Diophantine dagger" by Yurovitsky, which was also discussed in
NMBRTHRY:
[url]http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0901&L=nmbrthry&T=0&F=&S=&P=271[/url][/quote]

 flouran 2009-12-12 18:48

[QUOTE=maxal;198609]I guess that was the same guy who asked similar questions to a number of members of NMBRTHRY mailing list.
[/QUOTE]
You are correct. He emailed me on August 11 saying that his name was Konstantin Smirnov and that he was a post-graduate student studying number theory in Moscow. Noam Elkies mentioned that he had been emailed this same question a dozen times from Mr. Smirnov. I shall cut off communication with Smirnov immediately. I'm glad I have a spam filter.

As a side-note: I have recently discussed the problem with Brian Conrad, Noam Elkies, and Bjorn Poonen, and they said there is no known method for proving that no solutions exist. However, there is most definitely a [B]high likelihood[/B] that no solutions exist.

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