Sum of a prime and a square (Q921041)

The main result of the paper is that the number of natural numbers \(n\leq N\) such that the equation \(n=p+m^ 2\) with p a prime has no solution is \(\ll N^{1-\gamma},\) where \(\gamma (>0)\) and the constant implied by \(``\ll ''\) are effectively calculable. This is an analogue of a well- known theorem of H. L. Montgomery and R. C. Vaughan concerning ``Goldbach exceptions''. The author points out an error in his earlier paper [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 1391-1423 (1981; Zbl 0478.10033)], which does not, however, matter now because the new result is anyway stronger than that asserted earlier. In addition, he mentions that his present result has also been obtained by \textit{A. I. Vinogradov} [Acta Arith. 46, 33-56 (1985; Zbl 0597.10048)], with an ineffective constant in \(``\ll ''\).