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AN EFFECTIVE BOUND FOR GENERALISED DIOPHANTINE m-TUPLES

Part of: Multiplicative number theory Diophantine equations

Published online by Cambridge University Press:  06 November 2023

SAUNAK BHATTACHARJEE Open the ORCID record for SAUNAK BHATTACHARJEE [Opens in a new window] ,
ANUP B. DIXIT Open the ORCID record for ANUP B. DIXIT [Opens in a new window]  and
DISHANT SAIKIA Open the ORCID record for DISHANT SAIKIA [Opens in a new window]
SAUNAK BHATTACHARJEE
Affiliation:
IISER Tirupati, C/O Sree Rama Engineering College, (Transit Campus), Tirupati, Andhra Pradesh 517507, India e-mail: saunakbhattacharjee@students.iisertirupati.ac.in
ANUP B. DIXIT*
Affiliation:
Institute of Mathematical Sciences (HBNI), CIT Campus, Taramani, Chennai, Tamil Nadu 600113, India
DISHANT SAIKIA
Affiliation:
Freie Universität Berlin, Kaiserswerther Str. 16-18, Berlin 14195, Germany e-mail: saikiadishant@gmail.com
*
e-mail: anupdixit@imsc.res.in
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Abstract

For $k\geq 2$ and a nonzero integer n, a generalised Diophantine m-tuple with property $D_k(n)$ is a set of m positive integers $S = \{a_1,a_2,\ldots , a_m\}$ such that $a_ia_j + n$ is a kth power for $1\leq i< j\leq m$. Define $M_k(n):= \text {sup}\{|S| : S$ having property $D_k(n)\}$. Dixit et al. [‘Generalised Diophantine m-tuples’, Proc. Amer. Math. Soc. 150(4) (2022), 1455–1465] proved that $M_k(n)=O(\log n)$, for a fixed k, as n varies. In this paper, we obtain effective upper bounds on $M_k(n)$. In particular, we show that for $k\geq 2$, $M_k(n) \leq 3\,\phi (k) \log n$ if n is sufficiently large compared to k.

Keywords

Diophantine m-tuplesGallagher’s sievegap principlequantitative Roth’s theorem

MSC classification

Primary: 11D45: Counting solutions of Diophantine equations
Secondary: 11D72: Equations in many variables 11N36: Applications of sieve methods

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 242 - 253
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The research of the second author is partially supported by the Inspire Faculty Fellowship. The research of the first and third authors was supported by a summer research program in IMSc, Chennai.

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