Are there any known nonslow methods for solving diophantine systems?
I can't find books of mathematics that appear methods explaining how to solve diophantine systems in a manner "not slow", e.g. not force brute enumeration.
Are there any known nonslow methods for solving diophantine systems? I can't find books of mathematics that appear methods explaining how to solve diophantine systems in a manner "not slow", e.g. not force brute enumeration. 

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You must not have looked very hard. This will get you started: Nigel P. Smart. The Algorithmic Resolution of Diophantine Equations. London Mathematical Society Student Texts 41. Cambridge University Press, 1998. However, as Igor and zroslav have mentioned, some problems are unsolvable, others believed to be very hard, so don't be surprised that there are no "easy" methods. 

