Problem 931

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?

Status

Open

No claim settles this problem.

Er76d: Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
Gu04: Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section B35 "Products of consecutive numbers with the same prime factors", printed p. 138, which states the question for $(m+1)\cdots(m+k)$ and $(n+1)\cdots(n+l)$ with $k\ge l\ge3$ , gives the examples $2\cdots10$ with $14\cdot15\cdot16$ and $48\cdot49\cdot50$ and $2\cdots12$ with $98\cdot99\cdot100$ , and records Erdős's conjecture that for $k=l\ge3$ this happens only finitely many times; for $k>l$ Guy states only the question. Guy cites Erdős, Amer. Math. Monthly 87 (1980), 391--392. Library home: Guy 2004.