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

Number Theory