Random Diophantine Equations in the Primes

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. 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.