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arXiv · 2507.09899

On pairs of consecutive sequences with the same radicals

Noah Lebowitz-Lockard
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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 $i < k$.

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Noah Lebowitz-Lockard. 2025-07-14. On pairs of consecutive sequences with the same radicals. https://arxiv.org/abs/2507.09899

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