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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 [email protected]
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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.

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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