Grimm's Conjecture and Smooth Numbers

Abstract

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\le i\le k$ so that $p_i|n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for $g(n)$ by relating its study to the distribution of smooth numbers. Standard conjectures concerning smooth numbers in short intervals imply $g(n)=O(n^{\epsilon })$ for any $\epsilon >0$. We also prove unconditionally that $g(n) =O(n^\al)$ with $0.45<\al <0.46$. The study of $g(n)$ and cognate functions has some interesting implications for gaps between consecutive primes.