Apart from problems that specifically have to do with Turing machines, like Simulate a Turing Machine with the given description, are there any problems that require Turing-complete potentially nonterminating recursion?For example, are there any known non-Turing-machine-related problems that can't be solved in Coq?
Are there any known problems that require potential nontermination to solve?
computability;turing machines
Given a system of polynomials with integer coefficients (Diophantine system), find a solution if there is one, or else run forever. Any program which does this must necessarily be partial because the set of solutions of Diophantine systems is a complete c.e. set, see Hilbert's tenth problem.