On the integers of the form 𝑝²+𝑏²+2ⁿ and 𝑏₁²+𝑏₂²+2^{𝑛²}

Pan, Hao; Zhang, Wei

doi:10.1090/S0025-5718-2011-02445-2

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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Math. Comp. 80 (2011), 1849-1864 Request permission

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 $\{b_1^2+b_2^2+2^{n^2}: b_1,b_2,n\in \mathbb {N}\}.$

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