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arXiv · 1306.0765

Grimm's Conjecture and Smooth Numbers

Shanta Laishram·Ram Murty
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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.

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Shanta Laishram, Ram Murty. 2013-06-04. Grimm's Conjecture and Smooth Numbers. https://doi.org/10.1307/mmj/1331222852

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