A polynomial analogue of the twin prime conjecture

Pollack, Paul

doi:10.1090/S0002-9939-08-09351-9

A polynomial analogue of the twin prime conjecture
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Proc. Amer. Math. Soc. 136 (2008), 3775-3784 Request permission

Abstract:

We consider the problem of counting the number of (not necessarily monic) ‘twin prime pairs’ $P, P+M \in \mathbf {F}_q[T]$ of degree $n$, where $M$ is a polynomial of degree $< n$. We formulate an asymptotic prediction for the number of such pairs as $q^n\to \infty$ and then prove an explicit estimate confirming the conjecture in those cases where $q$ is large compared with $n^2$. 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 and Mullen.

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