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We use Vaughan's variation on Vinogradov's three-primes theorem to proveZariski-density of prime points in several infinite families of hypersurfaces,including level sets of some quadratic forms, the Permanent polynomial, and thedefining polynomials of some pre-homogeneous vector spaces. Three of thesefamilies are instances of a conjecture by Bourgain, Gamburd and Sarnakregarding prime points in orbits of simple algebraic groups. Our approach isbased on the formulation of a general condition on the defining polynomial of ahypersurface, which suffices to guarantee that Zariski-density of prime pointsis equivalent to the existence of an odd point.

Prime Points in Orbits: Some Instances of the Bourgain-Gamburd-Sarnak Conjecture