On a conjecture of Erdos, Graham and Spencer

JOURNAL OF NUMBER THEORY

卷:119

Article

Chen, Yong-Gao

关键词: Erdos-Graham-Spencer conjecture;    Erdos problem;    partition;    prime;   

DOI  :  10.1016/j.jnt.2005.11.003

来源: Elsevier

【 摘 要 】

It is conjectured by Erdos, Graham and Spencer that if 1 <= a(1) <= a(2) <= . . . <= a(s) with Sigma(s)(i=1) 1/a(i) < n - 1/30, then this sum can be decomposed into n parts so that all partial sums are <= 1. This is not true for Sigma(s)(i=1) 1/a(i) = n - 1/30 as shown by a(1) = 2, a(2) = a(3) = 3, a(4) = . . . = a(5n-3) = 5. In 1997, Sandor proved that Erdos-Graham-Spencer conjecture is true for Sigma(s)(i=1) 1/a(i) <= n - 1/2. In this paper, we reduce Erdos-Graham-Spencer conjecture to finite calculations and prove that Erdos-Graham-Spencer conjecture is true for Sigma(s)(i=1) 1/a(i) <= n - 1/3. Furthermore, it is proved that Erdos-Graham-Spencer conjecture is true if Sigma(s)(i=1) 1/a(i) < n - 1 / (log n + log log n - 2) and no partial sum (certainly not a single term) is the inverse of an positive integer. (c) 2005 Elsevier Inc. All rights reserved.

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