On $k$-th power Diophantine triples of the form $\{a^k, b, c\}$

TL;DR

This paper proves that for any integer power $k \\geq 3$, there are no Diophantine triples of the form \{a^k, b, c\} with increasing positive integers.

Contribution

It establishes a non-existence result for a specific class of $k$-th power Diophantine triples when $k \\geq 3$.

Findings

No such triples exist for $k \\geq 3$ with $1<a^k<b<c$.

The result narrows the search for $k$-th power Diophantine triples.

The proof covers all cases with $k \\geq 3$.

Abstract

In this paper, we prove that there are no $k$-th power Diophantine triples of the form $\{a^k,b,c\}$ for $k\geq 3$ and $1<a^k<b<c$.