Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-98-04414-1
MathSciNet review: 1452833
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $\theta \ge {\frac{99}{100}}$.


References [Enhancements On Off] (What's this?)

  • 1. W.J. Ellison and M. Mendès-France, Les nombres premiers, Hermann, 1974. MR 54:5138
  • 2. H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London 1974. MR 54:12689
  • 3. M.N. Huxley and H. Iwaniec, ``Bombieri's theorem in short intervals'', Mathematika 22 (1975), 188-194. MR 52:10620
  • 4. H. Iwaniec, ``The half dimensional sieve'', Acta Arith. 29 (1976), 69-95. MR 54:261
  • 5. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, Cambridge, 1995. MR 97e:11005b
  • 6. J. Wu, ``Théorèmes généralisés de Bombieri-Vinogradov dans les petits intervalles'', Quart. J. Math. Oxford (2), 44 (1993), 109-128. MR 93m:10090
  • 7. J. Wu, ``Distribution des nombres ${\mathcal{B}}$-libres dans les petits intervalles'', J. Théorie des Nombres de Bordeaux 5 (1993), 151-163. MR 94m:11125
  • 8. J. Wu, ``Sur l'équation $p+2=P_{2}$ dans les petits intervalles'', J. London Math. Soc. (2) 49 (1994), 61-72. MR 94k:11118

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11N05, 11N36

Retrieve articles in all journals with MSC (1991): 11N05, 11N36


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: https://doi.org/10.1090/S0002-9939-98-04414-1
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

American Mathematical Society