On Regular Higher Power Rational Diophantine Triples

A rational Diophantine $m$-tuple is a set $\{a_1,\dots ,a_m\}$ of distinct nonzero rational numbers such that $a_i{a}_j+1$ is a square for all $1\le i<j\le m$. Similarly, we may ask when $a_i{a}_j+1$ is a $k$-th power. Here, we study the case $k=4$ and produce some non-trivial infinite families of such triples. We show that there are infinitely many triples with positive elements for $k=4$. We also briefly consider the $k=6$ (sextic) and $k=8$ (octic) cases, explaining the difficulties in extending the method to higher exponents.