Primes, products and polynomials

摘要

As Conjecture F in the third paper of their celebrated series published under the rubric "Partitio Numerorum", Hardy and Littlewood propose an asymptotic estimate for the number of primes of the form ax~2 + bx + c with x integral and beneath a given bound, [6]. We might deepen the question implicit in their conjecture and require x to assume only prime values. For example, are there infinitely many primes of the form 2q~2 + 5 with q prime?