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Given a Diophantine equation it is not decidable if it has integer solution. I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties every member in $\mathcal D_{unique}$ is a ...

Is there a short Diophantine definition of the sum-of-divisors function? Is there a polynomial $p$ such that $$c = \sum_{d|n}d \ \leftrightarrow \ \exists x_1, \ldots x_{100}\ p(c,n,x_1, \ldots x_{...