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On the integers of the form $ p^2+b^2+2^n$ and $ b_1^2+b_2^2+2^{n^2}$


Authors: Hao Pan and Wei Zhang
Journal: Math. Comp. 80 (2011), 1849-1864
MSC (2010): Primary 11P32; Secondary 11A07, 11B05, 11B25, 11N36
DOI: https://doi.org/10.1090/S0025-5718-2011-02445-2
Published electronically: February 25, 2011
MathSciNet review: 2785483
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Abstract: We prove that the sumset $ \{p^2+b^2+2^n: p$ is prime and $ b,n\in\mathbb{N}\} $ has positive lower density. We also construct a residue class with an odd modulus that contains no integer of the form $ p^2+b^2+2^n$. Similar results are established for the sumset $ \{b_1^2+b_2^2+2^{n^2}: b_1,b_2,n\in\mathbb{N}\}. $


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Additional Information

Hao Pan
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: haopan79@yahoo.com.cn

Wei Zhang
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zhangwei_07@yahoo.com.cn

DOI: https://doi.org/10.1090/S0025-5718-2011-02445-2
Keywords: Positive lower density, arithmetical progression, prime, square, power of 2
Received by editor(s): May 23, 2009
Received by editor(s) in revised form: April 25, 2010
Published electronically: February 25, 2011
Additional Notes: The first author is supported by the National Natural Science Foundation of China (Grant No. 10771135 and 10901078).
Article copyright: © Copyright 2011 American Mathematical Society

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