Prime-power Diophantine tuples

Source abstract

A positive $D(n)$-$m$-tuple is a set $A=\{a_1,\ldots,a_m\}$ of distinct positive integers such that $a_i a_j+n$ is a square for every $i\ne j$. In 2005, Dujella and Luca obtained an absolute bound for the cardinality of a $D(p)$- or $D(-p)$-tuple of positive integers, uniformly in the prime $p$. We extend the underlying gap argument to prime powers. An explicit toric elimination certificate, proved by elementary linear algebra, replaces the unsaturated homogeneous elimination step and is valid modulo every prime power. Explicit degree and height bounds for this eliminant yield a uniform gap principle. Combined with the general bound for bounded $|n|$, this shows that every positive reduced $D(\pm p^r)$-tuple (i.e. tuple with elements not divisible by $p$) has less than $2^{121}$ elements, independently of $p$ and $r$. Thus, every positive $D(\pm p)$-tuple has at most $2^{121}$ elements, positive $D(\pm p^2)$-tuples have less than $2^{122}$ elements, and the maximal cardinality of a positive $D(\pm p^r)$-tuple is $O(r)$ uniformly in $p$.