Problem 849

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Statement. Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that

$$\binom{n}{k}=a$$

(with $1\leq k\leq n/2$) has exactly $t$ solutions?

Status. Open.

Source. erdosproblems.com/849, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #849, https://www.erdosproblems.com/849.

References.

• [Er96b] Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.
• [MRSTT22] Matomäki, Kaisa and Radziwiłł, Maksym and Shao, Xuancheng and Tao, Terence and Teräväinen, Joni, Singmaster's conjecture in the interior of Pascal's triangle. Q. J. Math. (2022), 1137-1177.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

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• matomaki_2022_singmaster_s_conjecture_interior_pascal_s