JOURNAL ARTICLE

Number of solutions to a special type of unit equations in two unknowns.

  • Published In: American Journal of Mathematics, 2024, v. 146, n. 2. P. 295 1 of 3

  • Database: Academic Search Ultimate 2 of 3

  • Authored By: Miyazaki, Takafumi; Pink, István 3 of 3

Abstract

For any fixed relatively prime positive integers a, b and c with min{a, b,c} > 1, we prove that the equation ax+by = cz has at most two solutions in positive integers x, y and z, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M. A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), no. 2, 897--922] which asserts that Pillai's type equation ax -by = c has at most two solutions in positive integers x and y for any fixed positive integers a, b and c with min{a,b} > 1. [ABSTRACT FROM AUTHOR]

Additional Information

  • Source:American Journal of Mathematics. 2024/04, Vol. 146, Issue 2, p295
  • Document Type:Article
  • Subject Area:Science
  • Publication Date:2024
  • ISSN:0002-9327
  • DOI:10.1353/ajm.2024.a923236
  • Accession Number:177796376
  • Copyright Statement:Copyright of American Journal of Mathematics is the property of Johns Hopkins University Press and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Looking to go deeper into this topic? Look for more articles on EBSCOhost.