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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A polynomial analogue of the twin prime conjecture
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by Paul Pollack PDF
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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Additional Information
  • Paul Pollack
  • Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
  • MR Author ID: 830585
  • Email: paul.pollack@dartmouth.edu
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3775-3784
  • MSC (2000): Primary 11T55; Secondary 11N32
  • DOI: https://doi.org/10.1090/S0002-9939-08-09351-9
  • MathSciNet review: 2425715