Abstract
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.
Publisher
Cambridge University Press (CUP)
Cited by
1 articles.
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1. Sets with the Property $$D(n)$$;Developments in Mathematics;2024