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Claim. P. Erdős and M. Rosenfeld, The factor-difference set of integers, Acta Arith. 79 (1997), no. 4, 353--359 (card), Proposition 3.2: for every positive integer $k$ there are integers $N_1<\cdots<N_k$ with $\lvert\bigcap_{i=1}^k D(N_i)\rvert\ge2$. The proof takes distinct odd primes $p_1,\ldots,p_n$, sets $2\alpha=p_1\cdots p_k+p_{k+1}\cdots p_n$ and $2\beta=p_1\cdots p_k-p_{k+1}\cdots p_n$, and turns each factorization of $p_1\cdots p_n$ into an integer $x^2-\alpha^2=y^2-\beta^2$ whose factor difference set contains $2\alpha$ and $2\beta$. With $k=2$ this answers Problem 885 yes for $k=2$. The paper also prints two triples, found by Barry Guiduli as the paper credits, whose sets share four values:

$$\{420,3780,14940,76860\}\subset D(6925500)\cap D(37901500)\cap D(108448956),$$ $$\{420,3780,61695,154332\}\subset D(2778300)\cap D(862552800)\cap D(5400442044).$$

All twenty-four memberships hold, since for each listed $d$ and $N$ the number $d^2+4N$ is a square of the parity of $d$. Either triple answers the problem yes for $k=3$. The year is the only date the journal record gives, so this page carries the first of January.

Covers. The instances $k=2$ and $k=3$.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Arithmetica (the paper thanks its referee). The site labels the problem OPEN, so its commentary crediting the paper is not acceptance.