On a conjecture of Graham
A well-known problem due to \textit{R. L. Graham} [Am. Math. Mon. 77, 775 (1970)] is the following: is it true that, for every chain \(0<a_ 1<...<a_ n\) of integers, there exist i,j such that \(\max_{i,j}(a_ i/(a_ i,a_ j))\geq n ?\) Let p(n) denote the largest prime number \(<2n\), and put \(f(n)=2n-p(n)\). The author shows that the answer to Graham's question is in the affirmative for all n for which \(f(n)<\sqrt{n}\). He then refines his argument to make use of Huxley's result \(f(n)\leq n^{7/12+\epsilon}\) for all sufficiently large n to confirm that Graham's conjecture is true when n is sufficiently big. [See \textit{M. Szegedy}, Combinatorica 6, 67-71 (1986; Zbl 0593.10002) for a proof of Graham's conjecture.]
scientific article; zbMATH DE number 4023365