On the integers of the form $p^2+b^2+2^n$ and $b_1^2+b_2^2+2^{n^2}$
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Abstract:
We prove that the sumset $\{p^2+b^2+2^n: p\text { is prime and } b,n\in \mathbb {N}\}$ 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 $\{b_1^2+b_2^2+2^{n^2}: b_1,b_2,n\in \mathbb {N}\}.$References
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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
- 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).
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2785483