paper

An effective bound on Generalized Diophantine m-tuples

arXiv:2209.15120

Abstract

For non-zero integers $n$ and $k \geq 2$ , a generalized Diophantine $m$ -tuple with property $D_{k} (n)$ is a set of $m$ positive integers $S = {a_{1}, a_{2}, \dots, a_{m}}$ such that $a_{i} a_{j} + n$ is a $k$ -th power for $1 \leq i < j \leq m$ . Define $M_{k} (n) := sup {∣ S ∣ : S$ has property $D_{k} (n)}$ . In a recent work, the second author, S. Kim and M. R. Murty proved that $M_{k} (n)$ is $O (lo g n)$ , for a fixed $k$ , as we vary $n$ . In this paper, we obtain effective upper bounds on $M_{k} (n)$ . In particular, we show that for $k \geq 2$ , $M_{k} (n) \leq 3 I ¨ (k) lo g n$ , if $n$ is sufficiently larger than $k$ .