A POLYNOMIAL ANALOGUE OF THE TWIN PRIME CONJECTURE

Abstract. We consider the problem of counting the number of (not neces-sarily monic) ‘twin prime pairs ’ P, P + M ∈ Fq [T] of degree n, where M is a polynomial of degree < n. We formulate an asymptotic prediction for the number of such pairs as qn → ∞ and then prove an explicit estimate confirm-ing the conjecture in those cases where q is large compared with n2. When M has degree n − 1, our theorem implies the validity of a result conditionally proved by Hayes in 1963. When M has degree zero, our theorem refines a result of Effinger, Hicks & Mullen. 1