Primes of the form 𝑝=1+𝑚²+𝑛² in short intervals

Wu, J.

doi:10.1090/S0002-9939-98-04414-1

Primes of the form $p=1+m^2 +n^2$ in short intervals
HTML articles powered by AMS MathViewer

by J. Wu PDF

Proc. Amer. Math. Soc. 126 (1998), 1-8 Request permission

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

William John Ellison, Les nombres premiers, Publications de l’Institut de Mathématique de l’Université de Nancago, No. IX, Hermann, Paris, 1975 (French). En collaboration avec Michel Mendès France. MR 0417077
H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
M. N. Huxley and H. Iwaniec, Bombieri’s theorem in short intervals, Mathematika 22 (1975), no. 2, 188–194. MR 389790, DOI 10.1112/S0025579300006069
H. Iwaniec, The half dimensional sieve, Acta Arith. 29 (1976), no. 1, 69–95. MR 412134, DOI 10.4064/aa-29-1-69-95
Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, 2nd ed., Cours Spécialisés [Specialized Courses], vol. 1, Société Mathématique de France, Paris, 1995 (French). MR 1366197
Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
Jie Wu, Distribution des nombres $\scr B$-libres dans de petits intervalles, J. Théor. Nombres Bordeaux 5 (1993), no. 1, 151–163 (French). MR 1251234, DOI 10.5802/jtnb.85
J. Wu, Sur l’équation $p+2=P_2$ dans les petits intervalles, J. London Math. Soc. (2) 49 (1994), no. 1, 61–72 (French). MR 1253011, DOI 10.1112/jlms/49.1.61