On sums of squares of primes and a k-th power of prime
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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.
KeywordsWaring–Goldbach problem Circle method Exceptional set
Mathematics Subject Classification11P32 11P55
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.
- 2.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)Google Scholar