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 and 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 can not 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 can not know whether or not it has a solution It only means that we can not 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