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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
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Abstract

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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.

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