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The first purpose of our paper is to show how Hooley's celebrated methodleading to his conditional proof of the Artin conjecture on primitive roots canbe combined with the Hardy-Littlewood circle method. We do so by studying thenumber of representations of an odd integer as a sum of three primes, all ofwhich have prescribed primitive roots.  The second purpose is to analyse the singular series. In particular, usingresults of Lenstra, Stevenhagen and Moree, we provide a partial factorisationas an Euler product and prove that this does not extend to a completefactorisation.

Vinogradov's three primes theorem with primes having given primitive roots