A NOTE ON ERDOS-STRAUS AND ERDOS-GRAHAM DIVISIBILITY PROBLEMS (WITH AN APPENDIX BY ANDRZEJ SCHINZEL)

摘要

In this paper we are interested in two problems stated in the book of Erdos and Graham. The first problem was stated by Erdos and Straus in the following way: Let n ∈ N_+ be fixed. Does there exist a positive integer κ such that κ∏i=1(n+i)∣κ∏i=1(n+κ+i)? The second problem is similar and was formulated by Erdos and Graham. It can be stated as follows: Can one show that for every nonnegative integer n there is an integer k such that n∏i=0(κ-i)∣(2κκ)? The aim of this paper is to give some computational results related to these problems. In particular we show that the first problem has positive answer for each n < 20. Similarly, we show the existence of desired n in the second problem for all n < 9. We also note some interesting connections between these two problems.