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
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by J. Wu

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

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

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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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