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Proceedings of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-08-09351-9
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
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.