Integers with consecutive divisors in small ratio. (English) Zbl 0543.10031

Vose, Michael D.

J. Number Theory 19, 233-238 (1984).

Let \(d_i\) run over the divisors of \(N\) and set \(F_c(N)=\sum(d_{i+1}/d_i-1)^c\) for \(c>1\). It was conjectured by Erdős that there exists an infinite sequence \(N_k\) for which \(F_2(N_ k)\) is bounded. This paper proves the conjecture, and more. A sequence \(N_k\) is constructed so that, for every \(c>1\), one has \(F_c(N_ k) \ll_c 1\) as \(k\to \infty\).
The same sequence \(N_k\) answers the following question of P. Erdős and M. V. Subbarao: Let \(f(N)\) denote the maximum of \(d_{i+1}-d_i\) for \(d_i<N^{\frac12}\). Is it true that, for any \(A>0\), there exist infinitely many \(N\) with \(f(N)\le N^{\frac12} (\log N)^{-A}\)? In fact one has \(f(N_k)\ll N_k^{\frac12} \exp(-c (\log N_k)^{\frac12})\) for a suitable constant \(c\).
The numbers \(N_k\) are of the form \(4^{ak^2} p_2^2 p_3^2\cdots p_k^2\), where \(a\) is a suitable fixed integer and the primes \(p_i\) are carefully chosen so that one has good control over the fractional part of \((\log p_i)/(\log 2)\).

Reviewer: D. R. Heath-Brown (Oxford)

MSC:

11N05	Distribution of primes
11A99	Elementary number theory
11B83	Special sequences and polynomials

Keywords:

elementary construction; distribution of divisors; small differences