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Consecutive tuples of multiplicatively dependent integers
Journal of Number Theory ( IF 0.7 ) Pub Date : 2021-08-24 , DOI: 10.1016/j.jnt.2021.07.021
Ingrid Vukusic 1 , Volker Ziegler 1
Affiliation  

This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are (2,8) and (3,9). For triples, we prove the following theorem: If a{2,8} is a fixed integer larger than 1, then there are only finitely many triples (a,b,c) of pairwise distinct integers larger than 1 such that (a,b,c), (a+1,b+1,c+1) and (a+2,b+2,c+2) are each multiplicatively dependent. Moreover, these triples can be determined effectively.



中文翻译:

乘法相关整数的连续元组

本文关注乘法依赖整数的连续对和连续三元组的存在。LeVeque 在 Pillai 方程上的一个定理意味着只有大于 1 的乘法相关整数的连续对是(2,8)(3,9). 对于三元组,我们证明以下定理:如果一种{2,8}是大于 1 的固定整数,则只有有限多个三元组(一种,b,C)成对的大于 1 的不同整数,使得(一种,b,C),(一种+1,b+1,C+1)(一种+2,b+2,C+2)都是乘法依赖的。此外,可以有效地确定这些三元组。

更新日期:2021-08-24
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