AN EFFECTIVE BOUND FOR GENERALISED DIOPHANTINE<i>m</i>-TUPLES

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 k th power for $1\leq i&lt; j\leq m$ . 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 .