Erdős Problem #850

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.

For a positive integer $u$, its prime support is $\operatorname{supp}_p(u)=\{p:p\mid u,\ p\text{ prime}\}$, with empty support for one. Equality of prime supports ignores exponents. The question requires $\operatorname{supp}_p(x+j)=\operatorname{supp}_p(y+j)$ simultaneously for $j=0,1,2$, with $x\ne y$. [S1] [S2]

Statement recorded in the supplied source.

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove unconditional nonexistence or exhibit x,y. [S1]

Short English statement

Can two distinct three-term consecutive blocks match term by term in their sets of distinct prime factors?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2339 (EP-850), 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 850, https://www.erdosproblems.com/850. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. Bibliographic information from the supplied record.

• S4 Makowski, Andrzej, On a problem of {E}rdős. Enseign. Math. (2) (1968), 193. Bibliographic information from the supplied record.

• S5 Shorey, Tarlok N. and Tijdeman, Rob, Arithmetic properties of blocks of consecutive integers. (2016), 455--471. Bibliographic information from the supplied record.