Problem 849

Erdős Problem #849: declared status 'open'. Formalized: yes. 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? Current best: There are no known examples for $t\geq 5$. Both Erd\H{o}s and Singmaster believed the answer to this question is no, and in fact that there exists an absolute upper bound on the number of solutions. Matomäki, Radziwill, Shao, Tao, and Teräväinen [MRSTT22] have proved that there are always at most two solutions if we restrict $k$ to$$k\geq \exp((\log n)^{2/3+\epsilon}),$$assuming $a$ is sufficiently large depending on $\epsilon>0$. Prize: no. OEIS: A003015, A003016, A059233, A090162, A098565, A180058, A182237. Tags: binomial coefficients, number theory.