Random Diophantine equations in the primes

Abstract We consider equations of the form where the variables are all taken to be 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 , , this holds for almost all such equations. This is based on work of Brüdern 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.