20091210, 05:36  #1  
Dec 2008
7^{2}×17 Posts 
Diophantine Equation
I recently received an email from some postgraduate student in Moscow (possibly a crank) who asked me the following question:
Quote:
Thanks! 

20091210, 11:59  #2 
Aug 2002
Buenos Aires, Argentina
53E_{16} Posts 
I received the same message at least 5 times in my email in several years, but I don't know how to solve it.

20091210, 22:32  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{3}·19·61 Posts 
No solutions with x<=y<=z<=1000.
(Bruteforced; 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. 
20091211, 03:51  #4 
Dec 2008
7^{2}×17 Posts 
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. 
20091211, 05:53  #5  
Aug 2006
2·11·271 Posts 
Quote:
1. It's not obvious how Matiyasevich's theorem (Hilbert X, RobinsonDavisPutnumMatiyasevich, 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 degree10 equation with only four variables. 3. Wiles' theorem doesn't seem wellequipped 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/ 

20091211, 06:11  #6  
Dec 2008
7^{2}·17 Posts 
Quote:
Would it be worth asking Wiles or one of his former Ph.D. students (i.e. Brian Conrad)? 

20091212, 06:17  #7  
Feb 2005
2^{2}×3^{2}×7 Posts 
Quote:
Here is some background info: Quote:
Last fiddled with by maxal on 20091212 at 06:20 

20091212, 18:48  #8  
Dec 2008
7^{2}·17 Posts 
Quote:
As a sidenote: 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 high likelihood that no solutions exist. Last fiddled with by flouran on 20091212 at 18:49 

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