Singmaster's Conjecture

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 $m>1$, put $N(m)=\#\{(n,k):n\geq0,\ 0\leq k\leq n,\ \binom nk=m\}$. Positions symmetric about a row's centre count separately, except that a central position is counted only once. The assertion is $\sup_{m>1}N(m)<\infty$. Values equal to one are excluded because they occur at infinitely many row ends. [S1] [S2]

Statement recorded in the supplied source.

Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Control the near-edge range, where the lower binomial index is small relative to the row, by an absolute constant. [S1]

Short English statement

Is there one universal bound on the number of positions in Pascal’s triangle occupied by any fixed value greater than one?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1132 (COMB-004), 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 K. Matomäki, M. Radziwiłł, X. Shao, T. Tao, and J. Teräväinen, Singmaster's Conjecture in the Interior of Pascal's Triangle, Quarterly Journal of Mathematics 73 (2022), 1137–1177. https://doi.org/10.1093/qmath/haac006. Bibliographic information from the supplied record.