Grimm's Conjecture and Smooth Numbers

TL;DR

This paper investigates the function g(n) related to Grimm's conjecture, establishing bounds by linking it to smooth numbers and providing both conditional and unconditional growth estimates, with implications for prime gaps.

Contribution

It provides new bounds for g(n) by connecting it to smooth number distribution and proves unconditional bounds with specific exponents, advancing understanding of Grimm's conjecture.

Findings

Conditional bounds: g(n) = O(n^ε) for any ε > 0.

Unconditional bounds: g(n) = O(n^α) with 0.45 < α < 0.46.

Implications for gaps between consecutive primes.

Abstract

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ 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.