The exceptional set for the sum of a prime and a square (Q5916457)

\textit{G. H. Hardy} and \textit{J. E. Littlewood} conjectured [Acta Math. 44, 1-70 (1923; JFM 48.0143.04)] that every large integer is either a square or the sum of a prime and a square. The authors show that the number of exceptional numbers not exceeding X for which this conjecture might be false is at most \(CX^{\theta}\), where \(C>0\) and \(\theta <1\) are explicitly computable. The methods used involve a significant extension of those used by \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arith. 27, 353-370 (1975; Zbl 0301.10043)] to discuss the analogous problem arising in connection with Goldbach's conjecture concerning sums of two primes. The authors mention that a claim by \textit{I. V. Polyakov} [Math. USSR., Izv. 19, 611-641 (1982); translation from Izv. Akad. Nauk SSR, Ser. Mat. 45, 1391-1423 (1982; Zbl 0478.10033)] to establish an asymptotic formula for the number of representations of almost all large numbers as a sum of a square and a prime cannot be substantiated, and that a different treatment of the present authors' result by \textit{A. I. Vinogradov} appeared in [Acta Arith. 46, 33-56 (1985; Zbl 0597.10048)].