Quantitative lower bound for Pythagorean pairs in boxes of prime powers

Rongzhong Xiao

For each positive integer $N$, let $p_i$ be the $i$-th prime for $i\in \{1,\dots ,N\}$, and set $${\Phi }_N=\left\{\prod _{i=1}^N{p}_i^{a_i}:a_i\in \{0,1,\dots ,N-1\}\right\}.$$ Write $f(…