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

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