, Volume 30, Issue 3, pp 443-446

On a problem of Erdős

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Abstract

Let s≥2 be an integer. Denote by f 1(s) the least integer so that every integer l>f 1(s) is the sum of s distinct primes. Erdős proved that f 1(s)<p 1+p 2+⋯+p s +Cslogs, where p i is the ith prime and C is an absolute constant. In this paper, we prove that f 1(s)=p 1+p 2+⋯+p s +(1+o(1))slogs=p 2+p 3+⋯+p s+1+o(slogs). This answers a question posed by P. Erdős.

The authors are supported by the National Natural Science Foundation of China, Grant Nos. 11126302, 11071121. J.-H. Fang is also supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions, Grant No. 11KJB110006 and the Foundation of Nanjing University of Information Science & Technology No. 20110421.