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 $n_2>2(n_1+k_1)$. It is not clear but it is possible that he meant to ask this question also permitting finitely many counterexamples. Indeed, without this caveat it is false - AlphaProof has found the counterexample
\[10! = 2^8\cdot 3^4\cdot 5^2\cdot 7\]
and
\[14\cdot 15\cdot 16 = 2^5\cdot 3\cdot 5\cdot 7,\]
so that $n_1=0$, $k_1=10$, $n_2=13$, and $k_2=3$.
See also [388].
This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
