  We consider equations of the form $a_{1}x_{1}^{k}+...+a_{s}x_{s}^{k}$ and
when they have solutions in the primes. We define an analogue of the Hasse
principle for solubility in the primes (which we call the prime Hasse
principle), and prove that, whenever $s\ge 3k+2$, this holds for almost all
such equations. This is based on work of Br\"udern and Dietmann on the Hasse
principle: arXiv:1212.4800. We then prove some further results about prime
solubility and the prime Hasse principle, including a partial converse, and
some counterexamples. Of particular interest are counterexamples of degree 2,
which show that the analogue of the Hasse-Minkowski theorem fails for prime
solubility.
