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AN EFFECTIVE BOUND FOR GENERALISED DIOPHANTINE m-TUPLES
Published online by Cambridge University Press: 06 November 2023
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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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