# Method(s) to turn a “single equation” problem into a “double equation” problem?

In the world of Diophantine analysis, there are single equation problems (e.g., “Find all solutions $$(x,y)$$ such that $$x^3=y^2+1$$”), double equation problems (e.g., “Find all numbers $$p$$ such that $$p+1$$ and $$p^2+1$$ are both the double of a square”), and problems involving three or more equations which share one or more of the variables.

What, if any, are the principal methods of [non-trivially] “upgrading” one type to a “higher” type? Can every single equation problem be [non-trivally] turned into a multiple-equation problem?

Note: By “non-trivially”, I mean not just rational transformations like writing an odd number $$x$$ as $$2u+1$$. Rather, I mean taking a single equation which [apparently] provides 'exactly $$N$$' units of information about the numbers/variables involved and turning it into two or more equations where the total amount of information is greater than '$$N$$'.