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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Primes of the form $p = 1 + m^{2} + n^{2}$ in short intervals


Author: J. Wu
Journal: Proc. Amer. Math. Soc. 126 (1998), 1-8
MSC (1991): Primary 11N05, 11N36
MathSciNet review: 1452833
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Abstract: In this note, we prove that for every $\theta \ge {\frac{115}{121}}$ and $x\ge x_{0}(\theta )$, the short interval $(x, x+x^{\theta }]$ contains at least one prime number of the form $p=1+m^{2}+n^{2}$ with $(m,n)=1$. This improves a similar result due to Huxley and Iwaniec, which requires $\theta \ge {\frac{99}{100}}$.


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Additional Information

J. Wu
Affiliation: Laboratoire de Mathématiques, Institut Elie Cartan – CNRS UMR 9973, Université Henri Poincaré (Nancy 1), 54506 Vandœuvre–lès–Nancy, France
Email: wujie@iecn.u-nancy.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04414-1
PII: S 0002-9939(98)04414-1
Keywords: Distribution of primes, applications of sieves methods
Received by editor(s): December 30, 1994
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1998 American Mathematical Society