An effective bound on Generalized Diophantine m-tuples

TL;DR

This paper establishes effective upper bounds on the size of generalized Diophantine m-tuples with property D_k(n), showing they grow at most logarithmically with n for fixed k, improving understanding of their limitations.

Contribution

The paper provides explicit upper bounds on M_k(n), demonstrating that the size of such tuples is bounded by a constant times log n for large n, which is a significant refinement over previous asymptotic results.

Findings

M_k(n) is bounded above by 3 * φ(k) * log n for large n

Effective bounds improve understanding of the structure of Diophantine m-tuples

The results apply for all k ≥ 2 and sufficiently large n

Abstract

For non-zero integers $n$ and $k\geq2$, a generalized 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 $1\leq i< j\leq m$. 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 $O(\log n)$, 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 $k\geq 2$, $M_k(n) \leq 3\,\phi(k)\, \log n$, if $n$ is sufficiently larger than $k$.