Erdős Problem #931

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

The lengths $k_1,k_2$ are fixed first. The two nonnegative starts may vary subject to disjointness. Equality concerns sets of distinct prime divisors, not equality of exponents or of the two products. [S1] [S2]

Statement recorded in the supplied source.

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? [S1]

Short English statement

For two fixed block lengths at least three, can only finitely many disjoint integer blocks have products involving exactly the same primes?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2387 (EP-931), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 931, https://www.erdosproblems.com/931. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (1976), 25-44. https://combinatorica.hu/~p_erdos/1976-39.pdf. Bibliographic information from the supplied record.

• S4 Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. Bibliographic information from the supplied record.