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ON QUOTIENTS OF VALUES OF EULER’S FUNCTION ON FACTORIALS

Part of: Multiplicative number theory Sequences and sets Elementary number theory

Published online by Cambridge University Press:  17 November 2021

AYAN NATH Open the ORCID record for AYAN NATH [Opens in a new window]  and
ABHISHEK JHA Open the ORCID record for ABHISHEK JHA [Opens in a new window]
AYAN NATH*
Affiliation:
Kaliabor College, Assam, India
ABHISHEK JHA
Affiliation:
Indraprastha Institute of Information Technology, New Delhi, India e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We investigate, for given positive integers a and b, the least positive integer $c=c(a,b)$ such that the quotient $\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$ is an integer. We derive results on the limit of $c(a,b)/(a+b)$ as a and b tend to infinity and show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ , with the exception of a set of density zero.

Keywords

Euler’s totient functionfactorialsdivisibility

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities 11N37: Asymptotic results on arithmetic functions 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 353 - 364
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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