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A generalization of a theorem of Sylvester and Schur

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Abstract

In this paper, we give an estimate of the lower bounds of the least common multiple of \(a,a+b,\ldots ,a+kb\) for \((a,b)=1,k\in \textit{N}^+\). Precisely, we prove that for any two coprime positive integers a and b, we have

$$\begin{aligned} L_{a,b,k}\ge \prod \limits _{p\mid b}p^{\text {Ord}_{p}^{k!}}\frac{1}{k!}\prod \limits _{i=0}^{k}(a+ib), \end{aligned}$$

where \(L_{a,b,k}\) is the least common multiple of \(a,a+b,\ldots ,a+kb\) and \(\text {Ord}_p^{k!}\) denotes the least s for which \(p^s\mid k!\) but \(p^{s+1}\not \mid k!\). In addition, we obtain a corollary that there is a number containing a prime divisor greater than k in the set \(\{a,a+b,\ldots ,a+kb\}\) for \((a,b)=1,b\ge 2\).

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Authors and Affiliations

  1. School of Computer Science and Artificial Intelligence, Aliyun School of Big Data, Changzhou University, Changzhou, 213164, China

    Hongguang Wu

  2. Department of Mathematics, Anhui University, Hefei, 230000, China

    JiaLe Sheng

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  1. Hongguang Wu
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  2. JiaLe Sheng
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Correspondence to Hongguang Wu.

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The first author is supported by the National Natural Science Foundation of China (Grants 11571166, 11631006, 11790272). The Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fundamental Research Funds for the Central Universities.

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Wu, H., Sheng, J. A generalization of a theorem of Sylvester and Schur. Ramanujan J 58, 131–144 (2022). https://doi.org/10.1007/s11139-021-00467-y

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  • DOI: https://doi.org/10.1007/s11139-021-00467-y

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