On a conjecture of Erdős, Graham and Spencer

✍ Scribed by Yong-Gao Chen

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 102 KB
• Volume: 119
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.11.003

✦ Synopsis

It is conjectured by Erdős, Graham and Spencer that if 1 a 1 a 2 • • • a s with 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 s i=1 1/a i = n -1/30 as shown by

In 1997, Sándor proved that Erdős-Graham-Spencer conjecture is true for s i=1 1/a i n -1/2. In this paper, we reduce Erdős-Graham-Spencer conjecture to finite calculations and prove that Erdős-Graham-Spencer conjecture is true for s i=1 1/a i n -1/3. Furthermore, it is proved that Erdős-Graham-Spencer conjecture is true if 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.