OPEN
Let $k_1\geq k_2\geq 3$. Are there only finitely many $n_2\geq n_1+k_1$ such that
\[\prod_{1\leq i\leq k_1}(n_1+i)\textrm{ and }\prod_{1\leq j\leq k_2}(n_2+j)\]
have the same prime factors?
Tijdeman gave the example
\[19,20,21,22\textrm{ and }54,55,56,57.\]
Erdős
[Er76d] was unsure of this conjecture, and thought perhaps if the two products have the same prime factors then $n_2>2(n_1+k_1)$, but could not even show that there must exist a prime between $n_1$ and $n_2$.
See also [388].
