Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T07:13:47.608Z Has data issue: false hasContentIssue false
  • English
  • Français

GAPS BETWEEN DIVISIBLE TERMS IN $a^{2}(a^{2}+1)$

Part of: Sequences and sets Elementary number theory

Published online by Cambridge University Press:  13 September 2019

TSZ HO CHAN Open the ORCID record for TSZ HO CHAN [Opens in a new window]
TSZ HO CHAN*
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA email thchan6174@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose $a^{2}(a^{2}+1)$ divides $b^{2}(b^{2}+1)$ with $b>a$. We improve a previous result and prove a gap principle, without any additional assumptions, namely $b\gg a(\log a)^{1/8}/(\log \log a)^{12}$. We also obtain $b\gg _{\unicode[STIX]{x1D716}}a^{15/14-\unicode[STIX]{x1D716}}$ under the abc conjecture.

Keywords

divisibilitygap principlehyperelliptic curveabc conjecture

MSC classification

Primary: 11B83: Special sequences and polynomials
Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 3 , June 2020 , pp. 396 - 400
Copyright
© 2019 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.)

References

Chan, T. H., ‘Common factors among pairs of consecutive integers’, Int. J. Number Theory 14(3) (2018), 871–880.Google Scholar
Chan, T. H., Choi, S. and Lam, P. C.-H., ‘Divisibility on the sequence of perfect squares minus one: the gap principle’, J. Number Theory 184 (2018), 473–484.Google Scholar
Voutier, P., ‘An upper bound for the size of integral solutions to Y m = f (X)’, J. Number Theory 53 (1995), 247–271.Google Scholar