Erdős Problem #388

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.

Use nonnegative integer starts $m_1,m_2$ and integer lengths $k_1,k_2\geq4$. The intervals of factors are $\{m_1+1,\ldots,m_1+k_1\}$ and $\{m_2+1,\ldots,m_2+k_2\}$. The displayed start inequality makes them disjoint. [S1] [S2]

Statement recorded in the supplied source.

Can one classify all solutions of $ \prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\leq m_2$? Are there only finitely many solutions? [S1]

Short English statement

Can all equal-product pairs of disjoint integer intervals, each longer than three terms, be classified, and are there only finitely many?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2090 (EP-388), 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 388, https://www.erdosproblems.com/388. Reference from the supplied record; the live problem page was not independently checked for this compilation.