Prime numbers of the form p = m 2 + n 2 + 1 in short intervals

where the summation runs over primes, a is a fixed non-zero integer and r(n) is the number of representations of n as a sum of two squares. This implies the first unconditional proof that there are infinitely many primes of the form p = m+n+1. Huxley and Iwaniec [1] considered primes of the formm+n+1 with (m,n) = 1 in the short interval (x, x+x]. They proved that for θ = 99/100 this interval contains primes of this type for every sufficiently large x and more precisely that the number of them is of the expected order of magnitude, that is x/(log x). Wu [7] improved this result to θ = 115/121 ≈ 0.9504. In this paper, we prove the following theorem. Theorem 1. For every θ ≥ 10/11 = 0.9090... and x ≥ x0(θ), we have ∑