Abstract
In this paper we prove that, with at most \( O{\left( {N^{{\frac{5} {{12}} + \in }} } \right)} \) exceptions, all positive odd integers n ≤ N with n ≡ 0 or 1(mod 3) can be written as a sum of a prime and two squares of primes.
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Hardy, G. H.: Littlewood, J . E.: Some problems of “partitio numerorum” III: On the expression of a number as a sum of primes. Acta Math., 44, 1–70 (1923)
Linnik, Yu.: Hardy–Littlewood problem on the representation as the sum of a prime and two squares. Dokl Akad Nauk SSSR., 124, 29–30 (1959)
Linnik, Yu.: An asymptotic formula in an additive problem of Hardy and Littlewood. Izv Akad Nauk SSSR, Ser Mat., 24, 629–706 (1960)
Liu, J. Y., Zhan, T.: Sums of five almost equal prime squares II. Sci. in China, Ser. A, 41, 710–722 (1998)
Wooley, T. D.: Slim exceptional sets for sums of four squares. Proc. London Math. Soc., 85, 1–21 (2002)
Ghosh, A.: The distribution of αp 2 modulo 1. Proc. London Math. Soc., 42, 252–269 (1981)
Leung, M. C., Liu, M. C.: On generalized quadratic equations in three prime variables. Mh. Math., 115, 133–169 (1993)
Prachar, K.: Primzahlverteilung, Springer, Berlin, 1957
Pan, C. D., Pan, C. B.: Fundamentals of analytic number theory, (in Chinese), Science Press, Beijing, 1991
Heath–Brown, D. R.: Prime numbers in short intervals and a generalized Vaughan’s identity. Canad. J. Math., 34, 1365–1377 (1982)
Titchmarsh, E. C.: The theory of the Riemann zeta–function, 2nd ed., University Press, Oxford, 1986
Liu, J. Y., Liu, M. C.: The exceptional set in the four prime squares problem. Illinois Journal of Mathematics, 44, 272–293 (2000)
Davenport, H.: Multiplicative number theory, 2nd., Springer, Berlin, 1980
Hua, L. K.: Introduction to Number Theory, Science press, Beijing, English version, Berlin, Heidelberg, New York, Springer, 1982
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Project supported by National Natural Science Foundation(No. 90304009) and Foundation of Qufu Normal University for Ph. D.
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Wang, M.Q., Meng, X.M. The Exceptional Set in the Two Prime Squares and a Prime Problem. Acta Math Sinica 22, 1329–1342 (2006). https://doi.org/10.1007/s10114-005-0701-7
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DOI: https://doi.org/10.1007/s10114-005-0701-7