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A polynomial equation with integral coefficients for which integral solutions are required.

‘Wolfram displays a table of some of the simplest possible Diophantine equations, distinguishing between those known to have integer solutions, those known to have no integer solutions, and those for which the question is still open.’

‘In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots.’

‘Turán goes on to say that Carl Siegel and Klaus Roth generalised the classes of Diophantine equations for which these conclusions would hold and even bounded the number of solutions.’

‘That conjecture offers a new way of expressing Diophantine problems, in effect translating an infinite number of Diophantine equations (including the equation of Fermat's last theorem) into a single mathematical statement.’

‘Instead of asking whether a given Diophantine equation has a solution, ask ‘for what equations do known methods yield the answer?’’