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**ABSTRACT:**In this paper, we shall prove that for a sufficiently large odd numberN, the equation $$N = p_1 + p_2 + p_3 ,\frac{N}{3} - N^{\tfrac{7}{{12}} + e}< p_i \leqslant \frac{N}{3} + N^{\tfrac{7}{{12}} + e} (i = 1,2,3)$$ has solutions.Acta Mathematica Sinica 01/1994; 10(4):369-387. · 0.48 Impact Factor -
##### Article: On the additive completion of primes

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**ABSTRACT:**Given a set B of positive integers, define S=S(B)={p+b: b∈B and p is a prime}. The aim is to find “thin” sets B such that, in a suitable sense, S contains almost all integers. Letting B(x) be the counting function of B, that is B(x)=|B∩[1,x]|, the author proves that for any ε>0 one can find a set B with B(x)=O(logx) and d ̲(S)≥1-ε, where d ̲ denotes the lower asymptotic density. Furthermore, for any function ω(x) with ω(x)→∞ as x→∞, there is a set B such that B(x)=O(ω(x)logx) and d(S)=1, where d is the natural density. Finally, if S satisfies x-S(x)≤x 1-logloglogx/loglogx for large x, then lim infB(x)/logx≥e γ . It should be noted that weaker upper estimates for B(x) were known [see e.g. M. Kolountzakis, Acta Arith. 77, 1-8 (1996; Zbl 0865.11066)] but lower estimates seem to be new. The bulk of the proof is a finite version of the theorem which is proved by means of a probabilistic argument.Acta Arithmetica 01/1998; 86(3). · 0.47 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**1. We prove that there is a set of nonnegative integers A, with counting function A(x) = #(A " [1Acta Arithmetica 10/1996; · 0.47 Impact Factor

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