On pairs of consecutive sequences with the same radicals

TL;DR

This paper uses the abc Conjecture to derive bounds on the length of consecutive integer sequences with the same radical, improving previous results and analyzing the frequency of such pairs.

Contribution

It provides new bounds on the maximum length of consecutive sequences with identical radicals and counts the number of such pairs, leveraging recent results on the abc Conjecture.

Findings

Bound on the maximum length of consecutive sequences with same radicals

Estimate of the number of pairs with same radicals within a range

Improved bounds compared to previous work

Abstract

Let $(m, n, k)$ be a tuple of integers with the property that if $i \leq k$, 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 $m < n \leq x$ 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 $i < k$.