Arithmetic progressions among powerful numbers

In this paper, we study $k$-term arithmetic progressions $N,N+d,\mathrm{.}\mathrm{.}\mathrm{.},N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d{\gg }_{\epsilon }{N}^{1\mathrm{/}2-\epsilon }$. On the other hand, there exist infinitely many $3$-term arithmetic progressions of powerful numbers with $d\ll N^{1\mathrm{/}2}$ unconditionally. We also prove some partial results when $k\ge 4$ and pose some open questions.