OPEN
Can one classify all solutions of
\[\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)\]
where $1<k_1<k_2$ and $m_1+k_1\leq m_2$? Are there only finitely many solutions?
More generally, if $k_1>2$ then for fixed $a$ and $b$
\[a\prod_{1\leq i\leq k_1}(m_1+i)=b\prod_{1\leq j\leq k_2}(m_2+j)\]
should have only a finite number of solutions.
See also [363] and [931].