== Algorithmic undecidability == 

 There's a dispute about the following text: 
 :On a different note — some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven that there is no Turing machine which can solve all Diophantine equations (see Hilbert's 10th problem). 
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 : There are thus some problems in number theory that will never be solved. We even know the shape of some of them, viz., Diophantine equations in nine variables; we simply do not know, and cannot know, which coefficients give us equations for which the following two statements are both true: there are no solutions, and we shall never know that there are no solutions. 
 The second assertion does not follow from the first.  Just because there is no Turing machine that can determine for every Diophantine equation whether or not it has a solution, it does not follow directly that there is a Diophantine equation for which we cannot know whether or not it has a solution.  It only means that we cannot know it simply by feeding it into a Turing machine.  It does not preclude our knowing it some other way.   
 A reference to a reliable source would be helpful here, such as   Reprinted in The Collected Works of Julia Robinson, Solomon Feferman, editor, pp.269–378, American Mathematical Society 1996.  In this, on page 326, we read it may be possible to devise ad hoc methods for particular equations.