Some Diophantine equations involving arithmetic functions and Bhargava factorials

2024/07/04 by Daniel M. Baczkowski, Saša Novaković, Baczkowski, Daniel M. +1 · 2 citations
Computer Science · Mathematics · #11A25 #11A41 #11D99 #Advanced Mathematical Identities #Advanced Mathematical Theories #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2407.03822

openalex publication_date 2024/07/04 · openalex created_date 2024/07/09 · openalex updated_date 2026/07/28

Abstract

F. Luca proved for any fixed rational number α>0 that the Diophantine equations of the form α m!=f(n!), where f is either the Euler function or the divisor sum function or the function counting the number of divisors, have only finitely many integer solutions (m,n). In this paper we generalize the mentioned result and show that Diophantine equations of the form α m₁!⋯ m_r!=f(n!) have finitely many integer solutions, too. In addition, we do so by including the case f is the sum of k\textsuperscriptth powers of divisors function. Moreover, we observe that the same holds by replacing some of the factorials with certain examples of Bhargava factorials.

Cited by

• Diophantine equations involving double factorials
• Diophantine equations involving powers of factorials

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