Another generalization of a theorem of Baker and Davenport

TL;DR

This paper extends a theorem related to Diophantine sets, proving new limitations on extending certain pairs to quintuples, especially when specific prime conditions are met.

Contribution

It provides a new generalization of Baker and Davenport's theorem, establishing conditions under which certain Diophantine pairs cannot be extended to quintuples.

Findings

The set {1, 3} cannot be extended to a Diophantine quintuple.

If b-1 is prime, the pair {1, b} cannot be extended to a Diophantine quintuple.

The main result generalizes previous work by Dujella and Pethő.

Abstract

Dujella and Peth\H{o}, generalizing a result of Baker and Davenport, proved that the set $\{1, 3\}$ cannot be extended to a Diophantine quintuple. As a consequence of our main result, it is shown that the Diophantine pair $\{1, b\}$ cannot be extended to a Diophantine quintuple if $b-1$ is a prime.