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A polynomial analogue of the twin prime conjecture

Author: Paul Pollack
Journal: Proc. Amer. Math. Soc. 136 (2008), 3775-3784
MSC (2000): Primary 11T55; Secondary 11N32
Published electronically: May 20, 2008
MathSciNet review: 2425715
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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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Additional Information

Paul Pollack
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755

Received by editor(s): July 10, 2007
Received by editor(s) in revised form: September 19, 2007
Published electronically: May 20, 2008
Additional Notes: The author was supported by an NSF Graduate Research Fellowship.
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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