On pairs of consecutive sequences with the same radicals

Abstract

Let $(m,n,k)$ be a tuple of integers with the property that if $i\le k$, then $m+i$ and $n+i$ have the same radical. Using a result on the abc Conjecture, we bound $k$ from above, improving a result of Balasubramanian, Shorey, and Waldschmidt. We also bound the number of pairs $(m,n)$ for which $m<n\le x$ and $m(m+1)\cdots (m+k-1))$ and $n(n+1)\cdots (n+\ell -1)$ have the same radical and the number of pairs for which $m+i$ and $n+i$ have the same radical for all $i<k$.