Abstract
In this short paper, we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of two squares of primes and a k-th power of prime for any integer \(k \ge 3\). Our results improve the recent results due to Brüdern (in: Sander, Steuding, Steuding (eds) From arithmetic to zeta-functions, Springer, Cham 2016). The similar method can be also applied to some related questions in this direction, and this can improve the previous results.
Similar content being viewed by others
References
Bourgain, J.: On the Vinogradov mean value. Trudy Mat. Inst. Steklova 296, 36–46 (2017)
Brüdern, J.: A ternary problem in additive prime number theory. In: Sander, J., Steuding, J., Steuding, R. (eds.) From Arithmetic to Zeta-Functions. Springer, Cham (2016)
Harman, G., Kumchev, A.: On sums of squares of primes. Math. Proc. Camb. Philos. Soc. 140, 1–13 (2006)
Harman, G., Kumchev, A.: On sums of squares of primes II. J. Number Theory 130, 1969–2002 (2010)
Hua, L.-K.: Some results in the additive prime number theory. Q. J. Math. (Oxford) 9, 68–80 (1938)
Hua, L.-K.: Additive Theory of Prime Numbers. Science Press, Rhode Island (1965)
Kumchev, A.: On Wyel sums over primes and almost primes. Mich. Math J. 54, 243–268 (2006)
Kumchev, A., Zhao, L.L.: On sums of four squares of primes. Mathematika 62(2), 348–361 (2016)
Leung, M.-C., Liu, M.-C.: On generalized quadratic equations in three prime variables. Monatsh. Math. 115(1–2), 133–167 (1993)
Li, T.Y.: On sums of squares of primes and a \(k\)th power of prime. Rocky Mt. J. Math. 42(1), 201–222 (2012)
Li, T.Y.: Enlarged major arcs in the Waring–Goldbach problem. Int. J. Number Theory 12, 205–217 (2016)
Liu, J.Y., Zhan, T.: Sums of five almost equal prime squares II. Sci. China Ser. A 41(7), 710–722 (1998)
Lü, G.S.: On sums of two squares of primes and a cube of a prime. Northeast Math. J. 19, 99–102 (2003)
Lü, G.S.: Note on a result of Hua. Adv. Math. 35, 343–349 (2006). (in Chinese)
Ren, X.M.: On expinential sum over primes and application in Waring–Goldbach problem. Sci. China Ser. A Math. 48, 785–797 (2005)
Schwarz, W.: Zur darstellung von zahlen durch summen von primzahlpotenzen II. J. Reine Angew. Math. 206, 78–112 (1961). (in Germany)
Wang, M.Q.: On the sum of a prime, the square of a prime and the k-th power of a prime. Indian J. Pure Appl. Math. 39, 251–271 (2008)
Wooley, T.D.: Slim exceptional sets for sums of four squares. Proc. Lond. Math. Soc. 85, 1–21 (2002)
Zhao, L.L.: On the Waring–Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. 108, 1593–1622 (2014)
Zhao, L.L.: The additive problem with one prime and two squares of primes. J. Number Theory 135, 8–27 (2014)
Acknowledgements
This work is supported by National Natural Foundation (No. 11301372), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130032120073). The authors would like to express their thanks to the referee for many useful suggestions and comments on the manuscript which led to an improvement of the original version of Lemma 2.5 and hence Theorems 1.1 and 1.2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
Rights and permissions
About this article
Cite this article
Liu, Z., Zhang, R. On sums of squares of primes and a k-th power of prime. Monatsh Math 188, 269–285 (2019). https://doi.org/10.1007/s00605-018-1181-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-018-1181-z