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All Random Solved Random Open
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. What if one just requires that the products have the same prime factors, say when $k_1=k_2$?