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Primes of the form in short intervals

Authors:	J Wu

Institution:	Laboratoire de Mathématiques, Institut Elie Cartan -- CNRS UMR 9973, Université Henri Poincaré (Nancy 1), 54506 Vandouvre--lès--Nancy, France

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 . This improves a similar result due to Huxley and Iwaniec, which requires $\theta \ge {\frac{99}{100}}$ .

Keywords:	Distribution of primes applications of sieves methods

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