Erdős Problem #930

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 intervals contain positive integers and are pairwise disjoint. A perfect power means $a^b$ with integers $a\geq1$ and $b\geq2$. The required minimum length depends only on the number of intervals, not on their positions. [S1] [S2]

Statement recorded in the supplied source.

Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then $ \prod_{1\leq i\leq r}\prod_{m\in I_i}m $ is not a perfect power? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: For each r>=2, prove that some sufficiently large common lower bound on the interval lengths rules out every perfect power, or construct counterexamples of unbounded length. [S1]

Short English statement

For each fixed number of disjoint integer intervals, is there a minimum block length that rules out a perfect power for their combined product?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2386 (EP-930), 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 930, https://www.erdosproblems.com/930. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Journal of Mathematics 19 (1975), 292-301. https://combinatorica.hu/~p_erdos/1975-46.pdf. Bibliographic information from the supplied record.

• S4 Mariusz Skałba, Products of disjoint blocks of consecutive integers which are powers, Colloquium Mathematicum 98 (2003), 1-3. https://eudml.org/doc/284443. Bibliographic information from the supplied record.