Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
Hostname: page-component-7ccbd9845f-zxw8g Total loading time: 0.246 Render date: 2023-01-30T10:49:55.535Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true
  • English
Bulletin of the Australian Mathematical Society Bulletin of the Australian Mathematical Society

Article contents

SPACINGS BETWEEN SELECTED PRIME DIVISORS OF AN INTEGER

Part of: Multiplicative number theory Elementary number theory

Published online by Cambridge University Press:  15 September 2022

JEAN-MARIE DE KONINCK Open the ORCID record for JEAN-MARIE DE KONINCK[Opens in a new window]  and
IMRE KÁTAI
JEAN-MARIE DE KONINCK*
Affiliation:
Department of Mathematics, Université Laval, Québec, Canada
IMRE KÁTAI
Affiliation:
Computer Algebra Department, Eötvös Loránd University, Budapest, Hungary e-mail: katai@inf.elte.hu
*
e-mail: jmdk@mat.ulaval.ca
Get access Rights & Permissions[Opens in a new window]

Abstract

Given a large integer n, determining the relative size of each of its prime divisors as well as the spacings between these prime divisors has been the focus of several studies. Here, we examine the spacings between particular types of prime divisors of n, such as prime divisors in certain congruence classes of primes and various other subsets of the set of prime numbers.

Keywords

prime divisors

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 13
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The work of the first author was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

References

De Koninck, J.-M. and Galambos, J., ‘The intermediate prime divisors of integers’, Proc. Amer. Math. Soc. 101(2) (1987), 213–216.CrossRefGoogle Scholar
De Koninck, J.-M. and Kátai, I., ‘Construction of normal numbers by classified prime divisors of integers’, Funct. Approx. Comment. Math. 45(2) (2011), 231–253.CrossRefGoogle Scholar
De Koninck, J.-M. and Kátai, I., ‘On the consecutive prime divisors of an integer’, Funct. Approx. Comment. Math. 66 (2022), 7–33.CrossRefGoogle Scholar
De Koninck, J.-M. and Kátai, I., ‘Consecutive neighbour spacings between the prime divisors of an integer’, Colloq. Math., to appear.Google Scholar
Drmota, M., Mauduit, C. and Rivat, J., ‘The sum-of-digits function of polynomial sequences’, J. Lond. Math. Soc. (2) 84(1) (2011), 81–102.CrossRefGoogle Scholar
Erdős, P., ‘On the distribution function of additive functions’, Ann. of Math. (2) 47 (1946), 1–20.CrossRefGoogle Scholar
Galambos, J., ‘The sequences of prime divisors of integers’, Acta Arith. 31(3) (1976), 213–218.CrossRefGoogle Scholar
Granville, A., ‘Prime divisors are Poisson distributed’, Int. J. Number Theory 3(1) (2007), 1–18.CrossRefGoogle Scholar
Granville, A., ‘Erratum: “Prime divisors are Poisson distributed”, Int. J. Number Theory 3(1) (2007), 1–18’, Int. J. Number Theory 3(4) (2007), 649–651.CrossRefGoogle Scholar
Harman, G., Prime-Detecting Sieves, London Mathematical Society Monographs Series, 33 (Princeton University Press, Princeton, NJ, 2007).Google Scholar
Mauduit, C. and Rivat, J., ‘Sur un problème de Gelfond: la somme des chiffres des nombres premiers’, Ann. of Math. (2) 171(3) (2010), 1591–1646.CrossRefGoogle Scholar
Shubin, A. V., ‘Fractional parts of noninteger powers of primes’, Math. Notes 108(3–4) (2020), 394–408.CrossRefGoogle Scholar
Vinogradov, I. M., ‘A general property of prime number distribution’, Mat. Sub. 7(49)(2) (1940), 365–372 (in Russian).Google Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

SPACINGS BETWEEN SELECTED PRIME DIVISORS OF AN INTEGER
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

SPACINGS BETWEEN SELECTED PRIME DIVISORS OF AN INTEGER
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

SPACINGS BETWEEN SELECTED PRIME DIVISORS OF AN INTEGER
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *