Problem 885

Erdős Problem #885: declared status 'open'. Formalized: yes. For integer $n\geq 1$ we define the factor difference set of $n$ by$$D(n) = \{\lvert a-b\rvert : n=ab\}.$$Is it true that, for every $k\geq 1$, there exist integers $N_1<\cdots<N_k$ such that$$\lvert \cap_i D(N_i)\rvert \geq k?$$ Prize: no. Tags: divisors, number theory.