Problem 931

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Statement. 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?

Formulation. The question is read, as Erdős's display (10) of 1976 reads it ("only finitely often", p. 29), and as the formal-conjectures statement states it, as asking whether, for fixed $k_1\ge k_2\ge3$, only finitely many pairs $(n_1,n_2)$ with $n_2\ge n_1+k_1$ occur. For a fixed $n_1$ finiteness is immediate: every term of the second block is composed of the primes dividing the first product, and Størmer's theorem leaves only finitely many pairs of consecutive such integers. The standing concerns the pairs reading.

Status. Open.

Source. erdosproblems.com/931, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #931, https://www.erdosproblems.com/931.

References.

• [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_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Current assessment

No current assessment is recorded. The status above is imported from the dated site record. The notes below record author-recorded transfers from research folders and are not independently reviewed. This page records no current literature search or independent assessment of proof coverage.

Known Results

The $S$-unit count of Lemma 3.2 of Pollack, Pomerance and Treviño (reconstruction) concerns pairs at a fixed difference, not products of blocks.

Linked library material

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